For the natural numbers m, n, if (1-y)^{m}(1+ | vedclass

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Question

$(1-y)^{m}(1+y)^{n}$

Coefficient of $y=1 .{ }^{n} C_{1}+{ }^{m} C_{1}(-1)$

$=n-m=10$ $\ldots(1)$

Coefficient of $\mathrm{y}^{2}\left(\mathrm{

Vedclass · Accepted Answer

$80$

Vedclass · Answer

$(1-y)^{m}(1+y)^{n}$

Coefficient of $y=1 .{ }^{n} C_{1}+{ }^{m} C_{1}(-1)$

$=n-m=10$ $\ldots(1)$

Coefficient of $\mathrm{y}^{2}\left(\mathrm{a}_{2}\right)$

$=1 .{ }^{n} \mathrm{C}_{2}-{ }^{n} \mathrm{C}_{1} \cdot{ }^{n} C_{1 .}+1 \cdot{ }^{m} C_{2}=10$

$=\frac{n(n-1)}{2}-m \cdot n+\frac{m(m-1)}{2}=10$

$m^{2}+n^{2}-2 m n-(n+m)=20$

$n+m=80$

$(n-m)^{2}-(n+m)=20$

By equation $(1)\,  \,(2)$

$\mathrm{m}=35, n=45$

For the natural numbers $m, n$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $a_{1}=a_{2}$ $=10$, then the value of $(m+n)$ is equal to:

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