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Distribution doubt

Angela has to distribute 15 chocolates among 5 of her children cupcake, pancake, lollipop, pie and gingerbread. She needs to make sure that cupcake get at least 3 and at most 6 chocolates. Find the number of ways can this be done. The answer is 435

Okay now I am getting 425. this is my way.
The question is same as asking the number of integer solutions of the equation
\(15=a+b+c+d+e\) constraint to the relation that \( 3\leq a \leq 6\) and \(1\leq b,c,d,e\leq15\). Which is equivalent to finding the coefficient of \(x^{15}\) in \((x^3+x^4+x^5+x^6)(x+x^2 + \cdots+x^{15})^4 \).

I checked from wolfram alpha that the coefficient is coming out to be 425. But the answer is 435. Can anyone provide a correct solution and tell me what I am doing wrong?

Note by Rishi Sharma
1 month ago

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You're not doing anything wrong. A brute force check using Python also reports that the answer is indeed 425. Prasun Biswas · 3 weeks, 6 days ago

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@Prasun Biswas Thanks. Then I guess there must have been some sort of printing mistake. Rishi Sharma · 3 weeks, 5 days ago

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