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I would like to ask this question, I am quite confused since a long time I haven't used this.

A king wants to distribute 10 identical coins among his 4 sons( every distribution is possible, it maybe that one son gets all the coins). He can do this in how many ways?

I am confused about the solution: $4^{10}$

or there is another solution: $n=10, m=4$, then combination of it is $C^{m-1}_{n+m-1}=C^{3}_{13}=286$ ways.

I think the second solution is correct but I don't understand why it has to minus $1$? And what different between identical coins and normal coins?

Hai Luu
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1 Answers1

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The intuition is to think of the divider bars in grocery stores. You use them to distinguish what belongs to which person. Generally to divide the groceries of $n$ people, you need $n-1$ divider bars, that's where the minus $1$ comes from. To get all the possible distributions of the coins, we take the coins and divider bars and consider all of their possible permutations: $(n+m-1)!$, but since the coins are identical, we don't care about their relative order, so we divide by $n!$ and the same goes for the divider bars - divide by $(m-1)!$.

In this picture you can see what it would look like if the first son got 2 coins, second son 4 coins, third only 1 coin and fourth 3 coins.

McDuck
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