Math / Discrete

QuestionTo play a lottery, five numbers must be selected from the numbers $1-51$ . Then, one additional number must be selected from the numbers 1 - 34. In how many ways are there to select numbers for this lottery? $\square$
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Studdy Solution

STEP 1

1. The order of selecting the five numbers from 1 to 51 does not matter.
2. The additional number is selected independently from the numbers 1 to 34.
3. There are no restrictions on the numbers selected other than the specified ranges.

STEP 2

1. Calculate the number of ways to select 5 numbers from 51.
2. Calculate the number of ways to select 1 number from 34.
3. Multiply the results from the previous steps to find the total number of ways to select the lottery numbers.

STEP 3

Calculate the number of ways to select 5 numbers from 51 using the combination formula:
$\binom{n}{r} = \frac{n!}{r!(n-r)!}$
For this problem, $n = 51$ and $r = 5$ :
$\binom{51}{5} = \frac{51!}{5!(51-5)!}$

STEP 4

Calculate the number of ways to select 1 number from 34. Since selecting 1 number from 34 is straightforward, it is simply:
$\binom{34}{1} = 34$

STEP 5

Multiply the results from STEP_1 and STEP_2 to find the total number of ways to select the lottery numbers:
$\text{Total Ways} = \binom{51}{5} \times \binom{34}{1}$
First, calculate $\binom{51}{5}$ :
$\binom{51}{5} = \frac{51 \times 50 \times 49 \times 48 \times 47}{5 \times 4 \times 3 \times 2 \times 1} = 2,869,685$
Then, multiply by the number of ways to select the additional number:
$\text{Total Ways} = 2,869,685 \times 34 = 97,569,290$
The total number of ways to select numbers for this lottery is:
$\boxed{97,569,290}$

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