Consider the data on x taking the values
0, 2, 4, 8,....., 2ⁿ with frequencies
ⁿC₀ , ⁿC₁ , ⁿC₂ ,...., ⁿCₙ respectively. If the
mean of this data is 728/2ⁿ, then n is equal to _________ .

Solution

Maths sir · 4 steps

The frequencies are binomial coefficients, so the weighted sum folds into (1+2)ⁿ by the binomial theorem.

1. Pair values and frequencies. The value 0 has frequency ⁿC₀, and for r = 1 to n the value 2ʳ has frequency ⁿCᵣ. The total frequency is Σᵣ₌₀ⁿ ⁿCᵣ = 2ⁿ.
2. Weighted sum. The value 0 adds nothing, so Σ f x = Σᵣ₌₁ⁿ ⁿCᵣ 2ʳ. The full sum from r = 0 is (1+2)ⁿ = 3ⁿ. Its r = 0 term is 1, so subtract it.

   x̄ = (Σ f x)/(Σ f) = (3ⁿ - 1)/2ⁿ

3. Match the mean. Set (3ⁿ - 1)/2ⁿ = 728/2ⁿ. Then 3ⁿ - 1 = 728, so 3ⁿ = 729 = 3⁶.
4. So. n = 6.

The idea to keep. When frequencies are ⁿCᵣ, the total frequency is 2ⁿ and Σ ⁿCᵣ aʳ = (1+a)ⁿ. Look for this pattern first, and watch which terms of the expansion are actually present.

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