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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?

Feb 10, 2023

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Divide it up by casework:

If the twins get 0 pieces - There are 10 pieces of candy and 3 children for 10 stars and 2 bars which makes \({12 \choose 2} = 66\) cases

If the twins get 1 piece - There are 8 pieces of candy and 3 children for 8 stars and 2 bars which makes \({10 \choose 2} = 45\) cases

If the twins get 2 pieces - There are 6 pieces of candy and 3 children for 6 stars and 2 bars which makes \({8 \choose 2} = 28\) cases

If the twins get 3 pieces - There are 4 pieces of candy and 3 children for 4 stars and 2 bars which makes \({6 \choose 2} = 15\) cases

If the twins get 4 pieces - There are 2 pieces of candy and 3 children for 2 stars and 2 bars which makes \({4 \choose 2} = 6\) cases

If the twins get 5 pieces - There are 0 pieces of candy and 3 children for 0 stars and 2 bars which makes \({2 \choose 2} = 1\) case

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

Feb 10, 2023