A woman has seven cookies - four chocolate chip and three oatmeal. She gives one cookie to each of her six children: Nicole, Ronit, Kim, Deborah, Mark, and Terrance. If Deborah will only eat the kind of cookie that Kim eats, in how many different ways can the cookies be distributed?
(A) 5040
(B) 50
(C) 25
(D) 15
(E) 12
OA D
Source: Manhattan Prep
A woman has seven cookies - four chocolate chip and three oatmeal. She gives one cookie to each of her six children
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Distributing 6 cookies can be done in the following ways:
3 chocolate, 3 oatmeal
4 chocolate, 2 oatmeal
Pairing Deborah and Kim (D and K) together, let's examine each way above:
D&K can each have a chocolate cookie, in which case there are 4 choices of children for the remaining chocolate cookie. The remaining children get oatmeal.
4 ways
Same logic holds if D&K get oatmeal cookies instead:
4 ways
Now let's examine the 4 chocolate/2 oatmeal scenario:
D&K get chocolate cookies. Two children can be selected from the remaining 4 to receive the other chocolate cookies as follows;
4!/2!2! = 6 ways. The other 2 children get oatmeal.
D&K get oatmeal cookies. The other children get chocolate. 1 way
Total ways 15,D
3 chocolate, 3 oatmeal
4 chocolate, 2 oatmeal
Pairing Deborah and Kim (D and K) together, let's examine each way above:
D&K can each have a chocolate cookie, in which case there are 4 choices of children for the remaining chocolate cookie. The remaining children get oatmeal.
4 ways
Same logic holds if D&K get oatmeal cookies instead:
4 ways
Now let's examine the 4 chocolate/2 oatmeal scenario:
D&K get chocolate cookies. Two children can be selected from the remaining 4 to receive the other chocolate cookies as follows;
4!/2!2! = 6 ways. The other 2 children get oatmeal.
D&K get oatmeal cookies. The other children get chocolate. 1 way
Total ways 15,D
