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.
+6252 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
NL=(8+4−14−1)=(113)=165NW=(7+4−14−1)=(103)=120N=NL⋅NW=19800
For (2) we have to find the sum of the cases for the amount the twins get
N0=(6+3−13−1)=(82)=28N1=(4+3−13−1)=(62)=15N2=(2+3−13−1)=(42)=6N3=11+6+15+28=50
