Let m be the smallest positive integer such that the coefficient of `x^2` in the expansion of `(1+x)^2 + (1 +x)^3 + (1 + x)^4 +........+ (1+x)^49 + (1 + mx)^50` is `(3n + 1) .^51C_3` for some positive integer `n.` Then the value of `n` is

Explanation

Coefficent of `x^(2)` in the expansion of
` {(1+x)^(2) +(1+x)^(3)+…..+(1+x)^(49)+(1+mx)^(50)} `
`implies ""^(2)C_(2) +""^(3)C_(2) +""^(4)C_(2) +…..+""^(49)C_(2) +""^(50)C_(2) *m^(2)=(3n+1)*""^(51)C_(3)`
`implies ""^(30)C_(3) +""^(50)C_(2)m^(2)=(3n+1)*""^(51)C_(3) " " [:. ""^(r )C_(r )+""^(r+1)C_(r) +....+""^(n)C_( r)=""^(n+1)C_(r+1)]`
`implies (50xx49xx48)/(3xx2xx1)+(50xx49)/(2)xxm^(2)=(3n+1)(51xx50s49)/(3xx2xx1)`
`implies m^(2)=51n+1`
`:. ` Minimum value of `m^(2)` for which (51n+1) is integer (perfect square ) for n=5 .
`:. m^(2)=51xx5+1implies m^(2)=256`
`:. m=16 and n=5`
Hence , the value of n is 5