The sum of all binomial coefficients in (a + b)ⁿ is x.

Solution

Maths sir · 3 steps

Put a = b = 1 and every term shrinks to its coefficient, so the sum is (1+1)ⁿ = 2ⁿ.

1. Set both to 1. In (a+b)ⁿ = Σᵣ₌₀ⁿ C(n, r) aⁿ⁻ʳ bʳ, setting a = b = 1 makes every power of a and b equal to 1. Each term is then just its binomial coefficient.

   (1+1)ⁿ = C(n, 0) + C(n, 1) + C(n, 2) + … + C(n, n)

2. Read off. The left side is 2ⁿ. So the sum of all the binomial coefficients is 2ⁿ, a power of 2 for every value of n.
3. So. The sum has the form xⁿ with base x = 2. The answer is 2.

The idea to keep. To add up all the coefficients of any expansion, put every variable equal to 1. For (a+b)ⁿ this gives 2ⁿ.

Answer from the NTA final answer key. Solution by PrepWiser.