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 :

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Binomial Theorem

For the natural numbers

m

n

, if

(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + \dots . + a_{m + n} y^{m + n}

and

a_{1} = a_{2} = 10

, then the value of

(m + n)

is equal to :

Answer: (C)

Expanding

(1 - y)^{m} (1 + y)^{n}

and comparing coefficients, the coefficient of

y

n - m

, so

n - m = 10

. The coefficient of

y^{2}

collects the three ways of reaching degree two:

(2 m) - mn + (2 n) = \frac{m ^{2} - m + n ^{2} - n - 2 mn}{2} = \frac{( n - m ) ^{2} - ( m + n )}{2}

Setting this equal to

10

gives

100 - (m + n) = 20

, so

m + n = 80

JEE Main · 2021 · 20 Jul 2021 · Shift 2 · Single correct option