Combinatorics

Problem 201

Find the probabilities for a 5-card hand from a 52-card deck:
a. Exactly 2 kings: $P(2 \text{ Kings})=0.0399$ b. All hearts: $P(\text{ All hearts})=0.000495$ c. Exactly 4 face cards: $P(4 \text{ Face Cards})=$

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Problem 202

How many distinct arrangements can be made with the letters in "ROBBERS"?

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Problem 203

How many unique arrangements can be made from the letters in "PROGRAMMING"? There are $\square$ distinct permutations.

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Problem 204

How many ways can six city commissioners be selected from fourteen candidates? Use the combination formula: $\binom{14}{6}$ .

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Problem 205

Find the number of distinct arrangements of the digits in $8,663,333$ .

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Problem 206

Find the number of committees of 2 teachers and 3 students from 6 teachers and 41 students. Answer: 3003.

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Problem 207

How many ways can you choose 5 books from 18 available books for your vacation? Use the formula for combinations: $\binom{18}{5}$ .

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Problem 208

How many ways can you form a committee of 3 teachers and 4 students from 9 teachers and 43 students?

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Problem 209

How many ways can you select all 48 items from a sample space without replacement? Express your answer in factorial notation.

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Problem 210

How many ways can 3 out of 12 bands be chosen for the top time slots? Use combinations: $C(12,3)$ .

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Problem 211

How many ways can you draw 100 unique names from a hat without replacement? Write your answer in factorial notation: $100!$

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Problem 212

How many unique patterns of 4 colors can be formed using the colors red, orange, yellow, green, blue, and violet?

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Problem 213

A scientist has discovered an organism that produces five offspring exactly one hour after its own birth, and then goes on to live for one week without producing any additional offspring. Each replicated organism also replicates at the same rate. At hour one, there is one organism. At hour two, there are five more organisms. How many total organisms are there at hour seven? 2,801 19,531 19,607 97,655

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Problem 214

6. Au banquet, tu vas chercher les entrées suivantes: 3 salades vertes, 2 salades césar et 3 soupes et tu vas les distribuer à 8 profs en espérant d'avoir une estampe pour ton passeport francophile. Si ton prof de Maths veut une soupe, combien de façons pourrais-tu distribuer ces huit entrées? A. 560 B. 5040 $3!2!3$ C. 210 D. 72
7. Du comité de Graduation (qui inclut 6 filles et 4 garçons), on doit choisir un groupe de 4 personnes avec au moins 3

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Problem 215

You flip two coins and roll a die. How many outcomes are possible? $\square$
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Problem 216

How many ways can you choose 2 different instructors from 6 for your statistics classes?

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Problem 217

How many ways can players choose 5 unique symbols from 9 total symbols for a password? Answer in simplest form: $\binom{9}{5}$ .

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Problem 218

How many ways can you choose 2 different instructors from 6 for your statistics classes? Use the formula for combinations: $\binom{6}{2}$ .

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Problem 219

How many unique 5-symbol passwords can be formed from 9 different symbols? Provide your answer in simplest form.

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Problem 220

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions?

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Problem 221

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions? Use the formula: $P(13, 5)$ .

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Problem 222

How many ways can nursing students choose 5 classes from 11 without repeating any class?

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Problem 223

How many ways can 6 people sit in 10 seats? Provide your answer in simplest form.

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Problem 224

How many ways can 35 new employees be grouped into sets of 5 without repetition? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

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Problem 225

How many ways can 11 out of 15 representatives be chosen for presentations? Calculate using combinations: $\binom{15}{11}$ .

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Problem 226

How many ways can 11 out of 15 representatives be chosen for presentations? Use the formula for combinations: $\binom{15}{11}$ .

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Problem 227

How many ways can a customer choose 13 different types of doughnuts from 21 available types? Use combinations: $\binom{21}{13}$ .

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Problem 228

Calculate the total combinations of a password with 3 digits ( $10$ options) and 8 letters ( $26$ options) without repetition.

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Problem 229

A bookseller selects 20 of 30 fiction and 10 of 25 nonfiction books. Calculate combinations using $C(n, k) = \frac{n!}{k!(n-k)!}$ .

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Problem 230

How many ways can you choose 2 characters from 8 volunteers for a play? Answer: $C(8, 2)$ .

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Problem 231

How many ways can 5 players be selected from 25 for a basketball game? Use the formula for combinations: $\binom{25}{5}$ .

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Problem 232

How many different car choices are there with 3 body styles, 2 colors, and 3 models? Calculate: $3 \times 2 \times 3$ .

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Problem 233

Calculate the number of possible passwords starting with "A" or "B", followed by 6 characters (digits or lower-case letters).

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Problem 234

How many ways can you choose 5 toppings from 11 options at a deli? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

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Problem 235

Calculate the number of ways to award gold, silver, and bronze medals to 8 sprinters in a 100-meter race.

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Problem 236

A pianist has 8 pieces for a recital. How many ways can she arrange them? Your answer is: $8!$

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Problem 237

In how many ways can 8 runners finish in 1st, 2nd, and 3rd places in an 800 meter race?

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Problem 238

How many ways can a student choose 3 classes from 8 needed classes to graduate? Use combinations: ${8 \choose 3}$ .

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Problem 239

How many ways can 5 runners finish a race without ties? Your answer is: $5!$

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Problem 240

Javier has 9 Judy Moody books. How many ways can he arrange them on his shelf? Answer: $9!$

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Problem 241

How many ways can you choose 3 students from a class of 22? Use the formula for combinations: $\binom{22}{3}$ .

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Problem 242

How many different pizzas can you create with 4 crusts, 3 cheeses, 6 meats, and 4 veggies if you choose 1 of each?

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Problem 243

Choose 4 toppings from 12. How many combinations are there? Answer: $\binom{12}{4}$

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Problem 244

A designer has 7 colors for a flag with 2 vertical stripes of different colors. How many flags can be made?

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Problem 245

How many ways can you choose 3 books from 9? Your answer is: $\binom{9}{3}$

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Problem 246

How many ways can you award first, second, and third prizes among 605 contestants? Your answer is: $605 \times 604 \times 603$

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Problem 247

How many ways can a person choose 4 different bags of chips from 12 varieties? Use the combination formula: $\binom{12}{4}$ .

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Problem 248

How many different outcomes are possible when rolling three different colored four-sided dice?

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Problem 249

How many gifts did 'True Love' receive on the 12 days of Christmas? Show your work using $n$ gifts on day $n$ .

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Problem 250

Avery can choose 1 math, 1 history, and 1 science class. How many different schedules are possible? Calculate: $4 \times 4 \times 3$ .

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Problem 251

How many car combinations can Aiden create with 5 paint colors, 2 wheels, 5 rims, and 4 window tints?

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Problem 252

Find the number of proper subsets of the set. $\{x \mid x$ is a day of the week $\}$ A. 256 B. 128 C. 64 D. 127

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Problem 253

How many different sequences can be formed with 5 identical process samples and 2 identical control samples?

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Problem 254

How many ways can you choose 5 chips from 140, containing exactly 1 nonconforming chip (10 total nonconforming)?

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Problem 255

How many ways can you select a sample of 5 chips from 140, containing exactly 1 nonconforming chip out of 10?

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Problem 256

f Ser... Studdy AI Pr... Learn Session Clos... (A) alyaqeensa... pose we want to choose 2 letters, without replacement, from the 5 letters $A, B, C, D$ , and $E$ . ecessary, consult a list of formulas.) a) How many ways can this be done, if the order of the choices is relevant? $\square$ (b) How many ways can this be done, if the order of the choices is not relevant? $\square$

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Problem 257

4 5 6 7 8 10 11 12 13 14 15 Esp
Selecting Council Members The presidents, vice presidents, and secretary-treasurers from each of four classes are eligible for an all-school council. Part: $0 / 2$
Part 1 of 2 (a) How many ways can four officers (president, vice president, secretary and treasurer) be chosen from these representatives if all representatives are eligible to become any of the four officers?
There are $\square$ ways they can be chosen.

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Problem 258

Choose 6 letters from 15 distinct letters. (a) Without order, how many ways? (b) With order, how many ways?

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Problem 259

Choose 4 objects from 18 distinct objects. How many ways can this be done if order doesn't matter? Use $C(18, 4)$ .

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Problem 260

Choose 6 colors from 10 distinct colors with order relevant. How many ways? Use $P(10, 6)$ .

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Problem 261

Choose 6 colors from 10 distinct colors. (a) How many ways if order matters? (b) How many ways if order doesn't matter?

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Problem 262

If 5 people shake hands once, how many handshakes occur? Generalize for $n$ people.

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Problem 263

Choose 6 colors from 10 distinct colors. If order matters, how many ways can this be done?

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Problem 264

Choose 6 colors from 10 distinct colors. (a) If order matters, how many ways? (b) If order doesn't matter, how many ways?

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Problem 265

Choose 5 letters from 17 distinct letters: (a) How many ways without order? (b) How many ways with order?

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Problem 266

Choose 5 letters from 17 distinct letters.
(a) Without order, how many ways? (b) With order, how many ways?
Answers: (a) $\binom{17}{5} = 6188$ (b) $17P5$

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Problem 267

Choose 7 from 10 distinct objects: (a) ways without order? (b) ways with order? Use $C(10,7)$ and $P(10,7)$ .

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Problem 268

Choose 6 colors from 8 distinct colors. (a) How many ways if order matters? (b) How many ways if order doesn't matter?

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Problem 269

If 5 people shake hands once, how many handshakes occur? Generalize for $n$ people.

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Problem 270

Meiosis I produces \_\_\_\_\_\_\_\_\_\_\_\_.
A. 2 diploid cells B. 2 haploid cells C. 4 daughter cells D. a parent and 2 daughter cells

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Problem 271

How many ways can Nick choose 5 pizza toppings from a menu of 8 toppings if each toppin can only be chosezonce?
Answer: $\qquad$

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Problem 272

Davon is picking out some movies to rent, and he has narrowed down his selections to 4 children's movies, 3 documentaries, 6 comedies, and 5 mysteries. How many different combinations of 9 movies can he rent if he wants all 6 comedies?
AnswerHow to enter your answer (opens in new window) 2 Points Keypad Keyboard Shortcuts

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Problem 273

How many different sums of money can you make with a penny, a nickel, a dime, and a quarter? *\$0.00 is a NOT consldered a sum of money. 15 7 12 11

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Problem 274

Find the minimum number of cars in a line such that one car is in front of four, one is behind four, and three are between two.

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Problem 275

How many different possible outcomes are there if you spin a spinner labeled Red, Blue, Yellow five times?

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Problem 276

How many unique ways can 5 friends sit in a row of 6 seats?

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Problem 277

How many unique 3-digit numbers with non-repeating digits are divisible by 5?

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Problem 278

Find the position of the word RUSE among the permutations of SURE, numbered 1 to 24 in alphabetical order.

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Problem 279

An ice cream store sells 2 drinks, in 3 sizes, and 4 flavors. In how many ways can a customer order a drink?
There are $\boxed{}$ ways that the customer can order a drink.
Question 1 Question 2 Question 3 Question 4 Question 5 Question 6

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Problem 280

11. Una scatola contiene 20 foglietti, numerati da 1 a 20. Dario ne estrae alcuni a caso, contemporaneamente. Quanti ne deve estrarre, come minimo, per essere certo di trovare almeno una coppia di numeri la cui somma faccia $22 ?$ (A) 12 (B) 10 (C) 9 (D) 11 (E) 13

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Problem 281

12. Ad una gara di cucina partecipano 6 cuochi, indicati con $A, B, C, D, E, F$ . Secondo le previsioni della vigilia, il cuoco $A$ è ritenuto più bravo di $B$ , il quale è più bravo di $C$ , che a sua volta è più bravo di D . Si ritiene inoltre che A sia più bravo anche di E. Se tutte le previsioni venissero rispettate, supponendo non possano esservi cuochi a pari merito, quanti sarebbero i possibili ordini d'arrivo della gara? (A) 21 (B) 24 (C) 18 (D) 20 (E) 16

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Problem 282

9. Tra tutti i numeri interi che vanno da 1000 a 3000 , quanti sono quelli nei quali cifra 7 compare esattamente una volta? (A) 512 (B) 576 (C) 600 (D) 486 (E) 450

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Problem 283

A person going to a party was asked to bring 2 different bags of chips. Going to the store, she finds 13 varieties.
How many different selections can she make? $\square$

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Problem 284

Does a trifecta bet in horse racing involve combinations or permutations? Explain your choice.

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Problem 285

Alice and Diego play a best of 5 chess match. What is the probability the match ends in 3 games? A. 0.50 B. 0.25 C. 0.20 D. 0.125

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Problem 286

A club has 3 women and 3 men. What is the probability of selecting 1 woman and 1 man for a committee? Round to 4 decimal places.

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Problem 287

What is the probability that 10 randomly arranged CDs out of 17 are in alphabetical order?

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Problem 288

Español
Lewis Middle School is going to select a committee. The committee will have a faculty member, a male student, a female student, a parent, and a school board member.
Here are the positions and the people interested in each. \begin{tabular}{|c|l|} \hline Position & \multicolumn{1}{|c|}{ People interested } \\ \hline Faculty member & Dr. Thompson, Mr. Smith \\ \hline Male student & Kareem, Joe, Bob \\ \hline Female student & Lucy, Ann, Ivanna \\ \hline Parent & Mrs. Green, Mr. Chang, Ms. Martinez \\ \hline School board member & Dr. Alexander, Mrs. Adams \\ \hline \end{tabular}
Based on this list, how many ways are there to fill the five committee positions? $\square$

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Problem 289

When making an ice cream sundae, you have a choice of 2 types of ice cream flavors: chocolate (C) or vanilla ( V ); a choice of 4 types of sauces: hot fudge ( H ), butterscotch (B), strawberry (S), or peanut butter (P); and a choice of 3 types of toppings: whipped cream (W), fruit (F), or nuts ( N ). If you are choosing only one of each, list the sample space in regard to the sundaes (combinations of ice cream flavors, sauces, and toppings) you could pick from.
Answer Tables Keypad Keyboard Shortcuts
Separate the elements of the sample space with commas. $\square$

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Problem 290

Part: $1 / 3$
Part 2 of 3
There are $\square$ items in the sample space.

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Problem 291

What is the probability that 10 randomly arranged CDs from your 17 CDs are in alphabetical order?

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Problem 292

3. There are 10 speakers at a conference with 5 of them men and different orders can they speak in if: a) Carly, David, and Elisa must immediately follow one another in any order? [2] b) Ben must go immediately after Alyssa? [2] c) They must present in alternating genders? [2] d) Jason and Shawn cannot speak immediately one after the other. [2] e) After the speeches, how many ways can they sit at a circular table for dinner? [2]

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Problem 293

A panel containing three on-off switches in a row is to be set. Now assume that no two adjacent switches can both be off. Explain why the fundamental counting principle does not apply. A. In the given situation, the fundamental counting principle does not apply because a multiple-part task does not satisfy the uniformity criterion. The number of possible settings for each switch does not depends on how previous switches were set. B. In the given situatior-the fundamental counting principle does not apply because a multiple-part task satisfies the uniformity criterion. The number of possible settings for each switch does not depend on how previous switches were set. C. In the given situation, the fundamental counting principle does not apply because a multiple-part task does not satisfy the uniformity criterion. After the first switch (two possibilities), other switches had either one or two possible settings depending on how previous switches were set. D. In the given situation, the fundamental counting principle does not apply because a multiple-part task satisfies the uniformity criterion. After the first switch (two possibilities), other switches had either one or two possible settings depending on how previous switches were set.

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Problem 294

TH-150-01CA) Braden Revermann 12/13/24 1:47 PM (?) HW Score: 78.57\%, 11 of 14 10.2 Question 12, 10.2.44 points Points: 0 of 1 Save
In how many ways could a club select two members, one to open their next meeting and one to close it, given that Alan will not be present?
N = \{Carl, Lisa, Jen, Adam, Alan, Tammy, Cathy, David, Sandy, Ashley\} $\square$ way(s) (Simplify your answer.)

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Problem 295

Meiosis is the process through which sex cells, or egg and sperm cells, are produced in multicellular organisms. This process is divided into two phases-meiosis I and meiosis II.
The picture below shows one stage that occurs during meiosis I.
What is the name of this phase and what occurs during this phase? A. Anaphase I; during this phase, homologous chromosome pairs separate, but sister chromatids remain attached to each other. B. Metaphase II; during this phase, homologous chromosomes line up in the middle of the cell. C. Anaphase II; during this phase, homologous chromosomes separate, and sister chromatids are pulled apart. D. Metaphase I; during this phase, homologous chromosome pairs line up in the middle of the cell.

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Problem 296

Sionplify the foumsing sows $\begin{array}{l} f(A, 3, c, D)=\operatorname{En}(2,3,12,13,14,15) \end{array}$

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Problem 297

A boy has a penny, nickel, dime, and quarter. What is the sample space for selecting 2 coins, and how many are there?

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Problem 298

Select 3 tires from 200. Each tire is ' $D$ ' (defective) or ' $N$ ' (non-defective). Find the sample space and total.

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Problem 299

List all combinations of picking 2 from Sam, Clyde, Will, Lisa, and Michelle. What is the total number of combinations?

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Problem 300

Probability Permutations and combinations: Problem type 2
Answer the questions below. (If necessary, consult a $\underline{\text { list of formulas.) }}$ (a) 74 athletes are running a race. A gold medal is to be given to the winner, a silver medal is to be given to the second-place finisher, and a bronze medal is to be given to the third-place finisher. Assume that there are no ties. In how many possible ways can the 3 medals be distributed? $\square$ (b) To log on to a certain computer account, the user must type in a 3-letter password. In such a password, no letter may be repeated, and only the lower case of a letter may be used. How many such 3-letter passwords are possible? (There are 26 letters in the alphabet.) $\square$

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1 2 3 4

Combinatorics

Problem 201

Problem 202

How many distinct arrangements can be made with the letters in "ROBBERS"?

Problem 203

How many unique arrangements can be made from the letters in "PROGRAMMING"? There are □\square□ distinct permutations.

Problem 204

How many ways can six city commissioners be selected from fourteen candidates? Use the combination formula: (146)\binom{14}{6}(614​).

Problem 205

Find the number of distinct arrangements of the digits in 8,663,3338,663,3338,663,333.

Problem 206

Find the number of committees of 2 teachers and 3 students from 6 teachers and 41 students. Answer: 3003.

Problem 207

How many ways can you choose 5 books from 18 available books for your vacation? Use the formula for combinations: (185)\binom{18}{5}(518​).

Problem 208

How many ways can you form a committee of 3 teachers and 4 students from 9 teachers and 43 students?

Problem 209

How many ways can you select all 48 items from a sample space without replacement? Express your answer in factorial notation.

Problem 210

How many ways can 3 out of 12 bands be chosen for the top time slots? Use combinations: C(12,3)C(12,3)C(12,3).

Problem 211

How many ways can you draw 100 unique names from a hat without replacement? Write your answer in factorial notation: 100!100!100!

Problem 212

How many unique patterns of 4 colors can be formed using the colors red, orange, yellow, green, blue, and violet?

Problem 213

Problem 214

Problem 215

You flip two coins and roll a die. How many outcomes are possible? □\square□Save answer

Problem 216

How many ways can you choose 2 different instructors from 6 for your statistics classes?

Problem 217

How many ways can players choose 5 unique symbols from 9 total symbols for a password? Answer in simplest form: (95)\binom{9}{5}(59​).

Problem 218

How many ways can you choose 2 different instructors from 6 for your statistics classes? Use the formula for combinations: (62)\binom{6}{2}(26​).

Problem 219

How many unique 5-symbol passwords can be formed from 9 different symbols? Provide your answer in simplest form.

Problem 220

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions?

Problem 221

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions? Use the formula: P(13,5)P(13, 5)P(13,5).

Problem 222

How many ways can nursing students choose 5 classes from 11 without repeating any class?

Problem 223

How many ways can 6 people sit in 10 seats? Provide your answer in simplest form.

Problem 224

How many ways can 35 new employees be grouped into sets of 5 without repetition? Use the formula for combinations: C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n-r)!}C(n,r)=r!(n−r)!n!​.

Problem 225

How many ways can 11 out of 15 representatives be chosen for presentations? Calculate using combinations: (1511)\binom{15}{11}(1115​).

Problem 226

How many ways can 11 out of 15 representatives be chosen for presentations? Use the formula for combinations: (1511)\binom{15}{11}(1115​).

Problem 227

How many ways can a customer choose 13 different types of doughnuts from 21 available types? Use combinations: (2113)\binom{21}{13}(1321​).

Problem 228

Calculate the total combinations of a password with 3 digits (101010 options) and 8 letters (262626 options) without repetition.

Problem 229

A bookseller selects 20 of 30 fiction and 10 of 25 nonfiction books. Calculate combinations using C(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n-k)!}C(n,k)=k!(n−k)!n!​.

Problem 230

How many ways can you choose 2 characters from 8 volunteers for a play? Answer: C(8,2)C(8, 2)C(8,2).

Problem 231

How many ways can 5 players be selected from 25 for a basketball game? Use the formula for combinations: (255)\binom{25}{5}(525​).

Problem 232

How many different car choices are there with 3 body styles, 2 colors, and 3 models? Calculate: 3×2×33 \times 2 \times 33×2×3.

Problem 233

Calculate the number of possible passwords starting with "A" or "B", followed by 6 characters (digits or lower-case letters).

Problem 234

How many ways can you choose 5 toppings from 11 options at a deli? Use the formula for combinations: C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n-r)!}C(n,r)=r!(n−r)!n!​.

Problem 235

Calculate the number of ways to award gold, silver, and bronze medals to 8 sprinters in a 100-meter race.

Problem 236

A pianist has 8 pieces for a recital. How many ways can she arrange them? Your answer is: 8!8!8!

Problem 237

In how many ways can 8 runners finish in 1st, 2nd, and 3rd places in an 800 meter race?

Problem 238

How many ways can a student choose 3 classes from 8 needed classes to graduate? Use combinations: (83){8 \choose 3}(38​).

Problem 239

How many ways can 5 runners finish a race without ties? Your answer is: 5!5!5!

Problem 240

Javier has 9 Judy Moody books. How many ways can he arrange them on his shelf? Answer: 9!9!9!

Problem 241

How many ways can you choose 3 students from a class of 22? Use the formula for combinations: (223)\binom{22}{3}(322​).

How many unique arrangements can be made from the letters in "PROGRAMMING"? There are $\square$ distinct permutations.

How many ways can six city commissioners be selected from fourteen candidates? Use the combination formula: $\binom{14}{6}$ .

Find the number of distinct arrangements of the digits in $8,663,333$ .

How many ways can you choose 5 books from 18 available books for your vacation? Use the formula for combinations: $\binom{18}{5}$ .

How many ways can 3 out of 12 bands be chosen for the top time slots? Use combinations: $C(12,3)$ .

How many ways can you draw 100 unique names from a hat without replacement? Write your answer in factorial notation: $100!$

You flip two coins and roll a die. How many outcomes are possible? $\square$
Save answer

How many ways can players choose 5 unique symbols from 9 total symbols for a password? Answer in simplest form: $\binom{9}{5}$ .

How many ways can you choose 2 different instructors from 6 for your statistics classes? Use the formula for combinations: $\binom{6}{2}$ .

How many ways can 13 players (7 girls, 6 boys) be assigned to 5 positions? Use the formula: $P(13, 5)$ .

How many ways can 35 new employees be grouped into sets of 5 without repetition? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

How many ways can 11 out of 15 representatives be chosen for presentations? Calculate using combinations: $\binom{15}{11}$ .

How many ways can 11 out of 15 representatives be chosen for presentations? Use the formula for combinations: $\binom{15}{11}$ .

How many ways can a customer choose 13 different types of doughnuts from 21 available types? Use combinations: $\binom{21}{13}$ .

Calculate the total combinations of a password with 3 digits ( $10$ options) and 8 letters ( $26$ options) without repetition.

A bookseller selects 20 of 30 fiction and 10 of 25 nonfiction books. Calculate combinations using $C(n, k) = \frac{n!}{k!(n-k)!}$ .

How many ways can you choose 2 characters from 8 volunteers for a play? Answer: $C(8, 2)$ .

How many ways can 5 players be selected from 25 for a basketball game? Use the formula for combinations: $\binom{25}{5}$ .

How many different car choices are there with 3 body styles, 2 colors, and 3 models? Calculate: $3 \times 2 \times 3$ .

How many ways can you choose 5 toppings from 11 options at a deli? Use the formula for combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$ .

A pianist has 8 pieces for a recital. How many ways can she arrange them? Your answer is: $8!$

How many ways can a student choose 3 classes from 8 needed classes to graduate? Use combinations: ${8 \choose 3}$ .

How many ways can 5 runners finish a race without ties? Your answer is: $5!$

Javier has 9 Judy Moody books. How many ways can he arrange them on his shelf? Answer: $9!$

How many ways can you choose 3 students from a class of 22? Use the formula for combinations: $\binom{22}{3}$ .

Choose 4 toppings from 12. How many combinations are there? Answer: $\binom{12}{4}$

How many ways can you choose 3 books from 9? Your answer is: $\binom{9}{3}$

How many ways can you award first, second, and third prizes among 605 contestants? Your answer is: $605 \times 604 \times 603$

How many ways can a person choose 4 different bags of chips from 12 varieties? Use the combination formula: $\binom{12}{4}$ .

How many gifts did 'True Love' receive on the 12 days of Christmas? Show your work using $n$ gifts on day $n$ .

Avery can choose 1 math, 1 history, and 1 science class. How many different schedules are possible? Calculate: $4 \times 4 \times 3$ .

Find the number of proper subsets of the set. $\{x \mid x$ is a day of the week $\}$ A. 256 B. 128 C. 64 D. 127

Choose 4 objects from 18 distinct objects. How many ways can this be done if order doesn't matter? Use $C(18, 4)$ .

Choose 6 colors from 10 distinct colors with order relevant. How many ways? Use $P(10, 6)$ .

If 5 people shake hands once, how many handshakes occur? Generalize for $n$ people.

Choose 5 letters from 17 distinct letters.
(a) Without order, how many ways? (b) With order, how many ways?
Answers: (a) $\binom{17}{5} = 6188$ (b) $17P5$

Choose 7 from 10 distinct objects: (a) ways without order? (b) ways with order? Use $C(10,7)$ and $P(10,7)$ .

If 5 people shake hands once, how many handshakes occur? Generalize for $n$ people.

Meiosis I produces \_\_\_\_\_\_\_\_\_\_\_\_.
A. 2 diploid cells B. 2 haploid cells C. 4 daughter cells D. a parent and 2 daughter cells

How many ways can Nick choose 5 pizza toppings from a menu of 8 toppings if each toppin can only be chosezonce?
Answer: $\qquad$