In Pascal's triangle, the sum of the coefficients in row n equals what?

• A. n
• B. n^2
• C. 2^n
• D. 2n

Answer: (C)

Pascal’s triangle row n contains the binomial coefficients C(n,0), C(n,1), ..., C(n,n). The sum of these coefficients is ∑{k=0}^n C(n,k). By the binomial theorem, (1 + 1)^n equals ∑{k=0}^n C(n,k)·1^{n−k}·1^k, which is just ∑_{k=0}^n C(n,k). Since (1 + 1)^n = 2^n, the sum of the row must be 2^n. Another way to see it: for each of n elements, you can either include it or exclude it in a subset, giving 2 choices per element, for a total of 2^n subsets, matching the row sum.