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. (5 pt.) If the coupons are identical, how many different ways are there to distribute the coupons so that at least one woman receives a coupon?
b. (5 pt.) If the coupons are different, how many different ways are there to distribute the coupons so that at least one woman receives a coupon?
My work
a) $\binom{20}{10}$ - $\binom{15}{10}$
b) $\binom{20}{10}$ - $\binom{5}{5}$$\binom{15}{5}$
Please verify my work. Thanks in advance
For part a, you've done it correctly.
For part b, you can start with your logic from part a in that you can distribute the coupons to the $20$ people in $\binom{20}{10}$ ways, but need to account for the fact that the coupons are different. Between the $10$ people who each received a coupon, we can order these coupons $10!$ ways, so the total number of ways to distribute the coupons in part b is $\binom{20}{10} * 10!$, or $20$ permute $10$.
Using a similar method, you can calculate the number of ways the coupons could be distributed among just the men and find your answer.