NDA Mathematics Question (2026) — Solution

Step-by-step solution

The number of elements in the set S is 2n+1. The number of subsets of S containing at most n elements is given by the sum: _ k=0 ^ n ^ 2n+1 C_ k We know the sum of all binomial coefficients is: _ k=0 ^ 2n+1 ^ 2n+1 C_ k = 2^ 2n+1 Using the symmetry property of binomial coefficients, ^ n C_ r = ^ n C_ n-r , we have: _ k=0 ^ n ^ 2n+1 C_ k = _ k=n+1 ^ 2n+1 ^ 2n+1 C_ k Therefore, the sum of the first half of the coefficients is exactly half of the total sum: _ k=0 ^ n ^ 2n+1 C_ k = 2^ 2n+1 2 = 2^ 2n Given that the number of such subsets is 1024, we can write: 2^ 2n = 1024 Since 1024 = 2^ 10 , we get: 2n = 10 n = 5 Answer: 5