Question

In the equation (x^2+y)^5, what is the coefficient of the term x^4y^3? what is the coefficient of the same term in the expansion of (3x^2+y)^5?

Answer 1

`\displaystyle (x+y)^n=\sum_(k=0)^n\binom{n}{k}x^(n-k)y^k`

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`\displaystyle (x^2+y)^n=\sum_(k=0)^n\binom{n}{k}x^(2n-2k)y^k\\ n=5\\ k=3\\\\\binom{5}{3}=(5!)/(3!2!)=(4\cdor5)/(2)=10`

It's 10.

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`\displaystyle (3x^2+y)^n=\sum_(k=0)^n\binom{n}{k}(3x)^(2n-2k)y^k=\sum_(k=0)^n\binom{n}{k}\cdot 3^(2n-2k)\cdot x^(2n-2k)y^k\\\\ n=5\\ k=4\\\\ \binom{5}{3}\cdot3^(2\cdot5-2\cdot4)=10\cdot3^(2)=10\cdot9=90`

It's 90