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# Counting

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Help how do you do this

There is a group of five children, where two of the children are twins.  How many ways can I distribute 10 identical pieces of candy to the children, if the twins must get an equal amount of candy?

Mar 13, 2023

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Let's call the twins A and B, and the other three children C, D, and E. We can approach this problem by breaking it down into cases:

Case 1: A and B each get 1 piece of candy
In this case, we have 8 pieces of candy left to distribute to 5 children (A, B, C, D, E) without any restrictions. This is equivalent to distributing 8 identical pieces of candy to 5 children, which can be done in (8 + 5 - 1) choose (5 - 1) = 462 ways using stars and bars.

Case 2: A and B each get 2 pieces of candy
In this case, we have 6 pieces of candy left to distribute to 3 children (C, D, E) without any restrictions. This is equivalent to distributing 6 identical pieces of candy to 3 children, which can be done in (6 + 3 - 1) choose (3 - 1) = 21 ways using stars and bars.

So the total number of ways to distribute the candy is 462 + 21 = 483. My Sutter Online Login

Mar 13, 2023