Combinations Identity

The sum of all binomial coefficients of a given order equals a power of two, a direct consequence of the binomial theorem central to JEE Advanced. Setting x = 1 in the expansion of (1 + x)^n gives the sum of C(n,k) for k from 0 to n equal to 2^n. For n = 8, this sum is 2^8 = 256. Combinatorially, the sum counts all subsets of an 8-element set, since C(8,k) counts subsets of size k, and the total number of subsets is 2^8. Option 128 = 2^7 uses the wrong exponent. Option 64 = 2^6 is too small. Option 512 = 2^9 is too large. Hence the sum is 256. Plausibility check: the identity says summing a full row of Pascal's triangle doubles the previous row's sum, and row 8 totals 2^8 = 256, matching both the subset count and the binomial substitution.