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a) Emma has 8 chocolate chip cookies and 7 gingerbread cookies. In how many ways can she distribute these cookies to four of her friends? (without breaking the cookies into pieces)
b) Jim has 6 cookies that he wants to distribute to 5 children. If all the cookies are identical, and two of the children are twins who insist on receiving an equal number of cookies, then how many ways can I distribute the cookies?
Focusing on the chocolate chip cookies, we notice that dividing 8 cookies into 4 groups is a classic stars/stripes problem. Distributing 8 cookies among 4 friends is C(11,3) ways.
In each of the above realities, you can also similarly divide 7 cookies into 4 groups to find C(10,3).
C(11,3)*C(10,3) = 165*120 = 19800.
Ways to distribute among 3 other children...
If twins get 0 cookies: C(8,2) = 28
1 cookie: C(6,2) = 15
2 cookies: C(4,2) = 6
3 cookies: I mean, the twins already have every cookie... (it's C(2,2) = 1)
Sum = 28 + 15 + 6 + 1 = 50 ways total.