How many ways and Why?

Probability Level 2

How many ways are there to arrange 3 chocolate chip cookies and 10 raspberry cheesecake cookies into a row of 13 cookies?


The answer is 286.

Mohammed Rizqallah by Mohammed Rizqallah , 23, Jordan

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1 solution

Jerry McKenzie
Dec 28, 2017

Arrange n=13 objects in which k 1 = 3 k_{1}=3 and k 2 = 10 k_{2}=10 are indistinguishable is given by:

n ! ∏ x ∈ X k x ! = 13 ! 3 ! ⋅ 10 ! = 13 ⋅ 12 ⋅ 11 ⋅ 10 ! 3 ⋅ 2 ⋅ 10 ! = 13 ⋅ 2 ⋅ 11 = 286 \Large \frac{n!}{ \prod_{x\in X}\enspace k_{x}!}=\frac{13!}{3!\cdot 10!}=\frac{13\cdot 12\cdot 11\cdot 10!}{3\cdot 2\cdot 10!}=13\cdot 2\cdot 11=286

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