The value of 1c1+3c3+5c5+7c7+ 7 where c0,c1,c2, cn are the binomial coefficients - Maths - Permutations and Combinations

Question

The value of 1c1+3c3+5c5+7c7+ 7 where c0,c1,c2, cn are the binomial
coefficients in the expansion of (1+x)to the power n ,is

C Reddy · Answer

given (1+x)^n = C0 + 1 C1 x +C2 x^2 + C3 x^3 +
differentiate on both sides, we get
n (1+x)^n-1 = C1 + 2 C2 x + 3 C3 x^2 +
substitute x=1,
n 2^(n-1) = C1 + 2 C2 + 3 C3 + (i)
substitute x= -1,
0 = C1 - 2 C2 + 3 C3 - (ii)
add (i) and (ii), we get
n 2^(n-1) = 2 C1 + 2 3 C3 +
n 2^(n-1)/2 = C1 + 3 C3 +
n 2^(n-2) = C1 + 3C3 + 5 C5 +

The value of 1c1+3c3+5c5+7c7+......7 where c0,c1,c2,... cn are the binomial coefficients in the expansion of (1+x)to the power n ,is