Question
The coefficient of λ^n μ^n in the expansion of [(1+λ)(1+μ)(λ+μ)]^n is ?
Joshi sir comment
(1+λ)n = C0 + C1x + ..................................Cnxn
similarly for (1+μ)n and
(λ+μ)n = C0λn + C1λn-1μ + ........................ + Cnμn
On multiplying these 3, we will get the coefficient of λnμn = ∑Cn3 (Here you should remember that C0 = Cn, C1 = Cn-1 and etc)
For getting the value of ∑Cn3 :
multiply the expansions of (1+x)n, (1-x)n and [1-(1/x2)]n and calculate the coefficient of x0, we get ∑Cn3
and if we multiply (1+x)n, (1-x)n and [1-(1/x2)]n, then we get (-1)n(1-x2)2n/x2n so coefficient of x0 in this expression will be (-1)n 2nCn (-1)n = 2nCn