If for positive integers r1, n2 , the coefficients of the (3 r)^ th and (r+2)^ th powers of x in the expansion of (1+x)^ 2 n are equal, then n is equal to:

Question

If for positive integers r>1, n>2 , the coefficients of the (3 r)^ th and (r+2)^ th powers of x in the expansion of (1+x)^ 2 n are equal, then n is equal to:

Quantrex Academy · Accepted Answer

Correct answer: A. 2 r+1

Expansion of (1+x)^ 2 n is 1+ ^ 2 n C ₁ x+ ^ 2 n C ₂ x^2+ .+ ^ 2 n C _r x^n+ ^ 2 n C _ r+1 x^ n+1 + + ^ 2 n C _ 2 n x^ 2 n As given ^ 2 n C _ r+2 = ^ 2 n C _ 3 r aligned & (2 n) ! (r+2) !(2 n-r-2) ! = (2 n) ! (3 r) !(2 n-3 r) ! & (3 r) !(2 n-3 r) !=(r+2) !(2 n-r-2) ! aligned Now, put value of n from the given choices. Choice (a) put n=2 r+1 in (1) LHS : (3 r) !(4 r+2-3 r) !=(3 r) !(r+2) ! RHS : (r+2) !(3 r) ! LHS = RHS

If for positive integers r>1, n>2 , the coefficients of the (3 r)^ th and (r+2)^ th powers of x in the expansion of (1+x)^ 2 n are equal, then n is equal to:

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