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# Counting

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Help I need 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?

Jan 24, 2023

#1
+2532
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Divide it up by cases and use stars and bars:

If the twins get 0 candies: There are 10 stars and 2 bars for \({12 \choose 2} = 66\) cases

If they get 1 candy: There are 8 stars and 2 bars for \({10 \choose 2} = 45\) cases

If they get 2 candies: there are 6 stars and 2 bars for \({8 \choose 2} = 28\) cases

If they get 3 candies: There are 4 stars and 2 bars for \({6 \choose 2} = 15\) cases

If they get 4 candies: There are 2 stars and 2 bars for \({4 \choose 2} = 6\) cases

If they get 5 candies: There are 0 stars and 2 bars for \({2 \choose 2} = 1\) case.

So, there are \(66 + 45 + 28 + 15 + 6 + 1 = \color{brown}\boxed{161}\) cases.

Jan 24, 2023