For natural numbers m, n if 1-ym1+yn=1+a1y+a2y2+…… and a1=a2=10 then (m, n) is

(1-y)m (1+y)n =1-my+c2 m y2+(-1)m cm ym m 1+ny+c2 n y2+....+cn yn nCofficient of y2 is c2 n-mn+c2 m -(1)Cofficient of y is n-m -(2)n-m=10, c2 n-mn+c2 m=10n(n-1)2+m(m-1)2-mn=10(m-n)2-(m+n)=20m+n=80 and n-m=102n=90n=45 and m=35(m, n)=(35, 45)

For natural numbers m, n if 1-ym1+yn=1+a1y+a2y2+…… and a1=a2=10 then (m, n) is

(1-y)m (1+y)n =1-my+c2 m y2+(-1)m cm ym m 1+ny+c2 n y2+....+cn yn nCofficient of y2 is c2 n-mn+c2 m -(1)Cofficient of y is n-m -(2)n-m=10, c2 n-mn+c2 m=10n(n-1)2+m(m-1)2-mn=10(m-n)2-(m+n)=20m+n=80 and n-m=102n=90n=45 and m=35(m, n)=(35, 45)

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For natural numbers m, n if ${(1 - y)}^{m} {(1 + y)}^{n} = 1 + a_{1} y + a_{2} y^{2} + \dots \dots$ and $a_{1} = a_{2} = 10$ then (m, n) is

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(20, 45)

(45,35)

(35,20)

(35,45)

answer is C.

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Detailed Solution

${(1 - y)}^{m} {(1 + y)}^{n} = (1 - my + {}^{m}{c_{2}} y^{2} + {(- 1)}^{m} {}^{m}{c_{m} y^{m}}) (1 + ny + {}^{n}{c_{2}} y^{2} + . . . . + {}^{n}{c_{n} y^{n}})$

Cofficient of $y^{2}$ is ${}^{n}{c_{2}} - mn + {}^{m}{c_{2}} - (1)$

Cofficient of y is $n - m - (2)$

$\begin{array}{rcl} n - m & = & 10, {}^{n}{c_{2}} - mn + {}^{m}{c_{2}} = 10 \\ \frac{n (n - 1)}{2} + \frac{m (m - 1)}{2} - mn & = & 10 \\ {(m - n)}^{2} - (m + n) & = & 20 \\ m + n & = & 80 and n - m = 10 \\ 2 n & = & 90 \\ n & = & 45 and m = 35 \\ (m, n) & = & (35, 45) \end{array}$