1. I have 8 pieces of lemon-flavored candy and 7 pieces of watermelon-flavored candy. In how many ways can I distribute this candy to four children?
2. I have 6 pieces of candy that I want to distribute to 5 children. If all the candy is identical, and two of the children are twins who insist on receiving an equal amount of candy, then how many ways can I distribute the candy?
Thanks a lot!
1. By stars and bars, the number of ways is C(10,4)*C(9,4) = 26460.
2. Using casework, there are C(7,3) + C(5,3) + C(3,3) = 35 + 10 + 1 = 46 ways to distribute the candies.
+6248 I don't agree with the answers given.
For (1) we can treat each candy individually and multiply the results.
Using stars and bars we can distribute 8 candies among 4 children as
\(N_L = \dbinom{8+4-1}{4-1} = \dbinom{11}{3} = 165\\ N_W=\dbinom{7+4-1}{4-1}=\dbinom{10}{3} = 120\\ N = N_L \cdot N_W = 19800\)
For (2) we have to find the sum of the cases for the amount the twins get
\(N_0 = \dbinom{6+3-1}{3-1} = \dbinom{8}{2} = 28\\ N_1 = \dbinom{4+3-1}{3-1} = \dbinom{6}{2} = 15\\ N_2 = \dbinom{2+3-1}{3-1} = \dbinom{4}{2} = 6\\ N_3 = 1\\ 1 + 6 + 15 + 28 = 50\)
