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?
2026-03-28 22:29:52.1774736992
What is the coefficient of $x^4y^3z^3$ in the expansion of $(5x+1y+5z)^{10}$?
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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
$$4200\cdot5^4\cdot1^3\cdot5^3=328125000$$