Question

10 coupons are given to 20 shoppers in a store. Each shopper can receive at most one coupon. 5 of the shoppers are women and 15 of the shoppers are men. (a) If the coupons are identical, how many ways are there to distribute the coupons so that at least one woman receives a coupon?

Answer 1

Answer:

$\, {20 \choose 10} - {15 \choose 10}$

Step-by-step explanation:

We can find the total amount of ways at least a woman receives a coupon by calculating the total amount of possibilities ot distribute the coupon and substract it to the total amount of possibilities to distribute 10 coupons to the 15 men (this is the complementary case that at least a woman receives a cupon).

- The total amount of possibilities to distribute the coupons among the 20 shoppers is equivalent to the total amount of ways to pick a subset of 10 elements from a set of 20. This is the combinatiorial number of 20 with 10, in other words, $\, {20 \choose 10} = \frac{20!}{10!10!} .$

- To calculate the total amount of possibilities to distribute the coupons among the 15 men, we need to make the same computation we made above but with a set of 15 elements instead of 20. This gives us  $\, {15 \choose 10} = \frac{15!}{10!5!}$ possibilities.

Therefore, we have $\, {20 \choose 10} - {15 \choose 10}$ possibilities to distribute the coupons so that at least one woman receives a coupon.

I hope that works for you!