# What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$?

What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$?

So, would I start by using the binomial or multinomial theorem? Not entirely sure where to start here?

• The coefficient of $a^4b^3c^3$ in the expansion of $(a+b+c)^{10}$ is $\frac{10!}{4!\,3!\,3!}=4200$, according to the multinomial theorem. – egreg Sep 19 '14 at 23:37
• @vvv123: The multinomial theorem is the quickest way, but you could certainly apply the binomial theorem twice to get the same result, by grouping. – Andrey Kaipov Sep 19 '14 at 23:42
• – Lucian Sep 20 '14 at 3:28

Write the product of ten copies of $(5x+1y+5z)$, and count the choices: you pick $4$ times an $x$ among the factors, $3$ times an $y$ in the remaining factors, and in the remaining $3$ times a $z$, hence the number of possibilities is:
$${10 \choose 4,3,3}={10\choose4}{10-4\choose3}{10-4-3\choose3}=210\cdot20\cdot1=4200$$
Each of these $4200$ possibilities contributes a factor $5^4\cdot1^3\cdot5^3$, hence the coefficient you are looking for is
$$4200\cdot5^4\cdot1^3\cdot5^3=328125000$$