For natural numbers m ,n ,if(1-y)^m(1+y)...

For natural numbers `m ,n ,if(1-y)^m(1+y)^n=1+a_1y+a_2y^2+... , and a_1=a_2=10 , then` a. `m < n` b. `m > n` c. `m+n=80` d. `m-n=20`

`m lt n`

`m gt n`

`m+n=80`

`m-n=20`

Text Solution

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The correct Answer is:

A, C

`(1-y)^(m)(1+y)^(n)`
`= (1-.^(m)C_(1)y+.^(m)C_(2)y^(2) - "……") (1+.^(n)C_(1)y+.^(n)C_(2)y^(2) + "…..")`
`= 1+(n-m)y+{(m(m-1))/(2) + (n(n-1))/(2)-mn} y^(2) + "….."`
Given,
`a_(1) = 10`
`rArr a_(1) = n - m = 10" " (1)`
`a_(2) = (m^(2) + n^(2) - m - n - 2mn)/(2) = 10`
`(m-n) = 80" "(2)`
Solving (1) and (2), we get `m = 35, n = 45`

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