Math  /  Discrete

QuestionPapa John's currently offers 16 different pizza toppings. Calculate how many different three-topping pizzas could you create with 16 unique toppings showing your work on the answer sheet.

Studdy Solution

STEP 1

What is this asking? How many 3-topping pizza combinations can we make from 16 available toppings? Watch out! The order of toppings doesn't matter (pepperoni, mushroom, onion is the same as onion, mushroom, pepperoni), so it's a combination, not a permutation!

STEP 2

1. Set up the combination formula
2. Calculate the combinations

STEP 3

We're dealing with **combinations**, not permutations, because the order of the toppings doesn't change the pizza.
If we had pepperoni, onions, and mushrooms, it's the same delicious pizza no matter how they arrange those toppings!
The formula for combinations is: C(n,k)=n!k!(nk)! C(n, k) = \frac{n!}{k!(n-k)!} Where nn is the **total number of items** to choose from, and kk is the **number we're choosing**.

STEP 4

In our case, n=16n = \textbf{16} (total toppings available) and k=3k = \textbf{3} (toppings we want on our pizza).
Let's plug those values into our formula: C(16,3)=16!3!(163)! C(16, 3) = \frac{16!}{3!(16-3)!}

STEP 5

Let's simplify that **factorial** in the denominator: C(16,3)=16!3!13! C(16, 3) = \frac{16!}{3!13!}

STEP 6

Now, let's expand the factorials and get ready to rock: C(16,3)=16151413121(321)(13121) C(16, 3) = \frac{16 \cdot 15 \cdot 14 \cdot 13 \cdot 12 \cdot \dots \cdot 1}{(3 \cdot 2 \cdot 1)(13 \cdot 12 \cdot \dots \cdot 1)}

STEP 7

Notice that we have 13!13! in both the numerator and denominator!
Let's divide to one: C(16,3)=161514321 C(16, 3) = \frac{16 \cdot 15 \cdot 14}{3 \cdot 2 \cdot 1}

STEP 8

We can simplify further.
Divide 1515 by 33 to get 55 and divide 1414 by 22 to get 77.
Divide 33 by 33 to get 11, 22 by 22 to get 11, and 11 stays as 11. C(16,3)=1657111 C(16, 3) = \frac{16 \cdot 5 \cdot 7}{1 \cdot 1 \cdot 1}

STEP 9

Multiply those numbers together: C(16,3)=1657=560 C(16, 3) = 16 \cdot 5 \cdot 7 = 560

STEP 10

There are 560\textbf{560} different three-topping pizza combinations we can create!

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