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?

#Combinatorics

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Comments

You're not doing anything wrong. A brute force check using Python also reports that the answer is indeed 425.

Prasun Biswas - 4 years, 7 months ago

Thanks. Then I guess there must have been some sort of printing mistake.

Rishi Sharma - 4 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$