For natural numbers m,n, if (1-y)^m(1+y)^n=1+a_1y+a_2y^2++a_m+ny^m+n and a_1=a_2=10, then …

Question

For natural numbers $m,n$, if $(1-y)^m(1+y)^n=1+a_1y+a_2y^2+\cdots+a_{m+n}y^{m+n}$ and $a_1=a_2=10$, then the value of $(m+n)$ is equal to

Accepted Answer

Expanding: coefficient of $y$: $n-m=10$. Coefficient of $y^2$: ${}^nC_2-mn+{}^mC_2=10$

$$\frac{n(n-1)}2+\frac{m(m-1)}2-mn=10\quad\Rightarrow\quad(m-n)^2-(m+n)=20$$

$$100-(m+n)=20\quad\Rightarrow\quad m+n=80$$

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