A set S contains (2n+1) elements. If the number of subsets of S which contain at most n elements is 1024 , then what is the value of n ?

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 k We know the sum of all binomial coefficients is: _ k=0 ^ 2n+1 2n+1 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 k = _ k=n+1 ^ 2n+1 2n+1 k Therefore, the sum of the first half of the coefficients is exactly half of the total sum: _ k=0 ^ n 2n+1 k = 2^ 2n+1 2 = 2^ 2n Given that the number of such subsets is 102