If Σₖ₌₁¹⁰ K²(10_(C_K))²=22000 L , then L is equal to ________.

Solution

Maths sir · 3 steps

Absorb each K with KC(10, K) = 10C(9, K-1), then it is a Vandermonde sum.

1. Absorb the K. Use kC(n, k) = nC(n-1, k-1) with n = 10. Then KC(10, K) = 10C(9, K-1), so K²C(10, K)² = 100C(9, K-1)².
2. Sum of squares. As K runs from 1 to 10, K-1 runs from 0 to 9. The sum of the squares of all the coefficients of (1+x)⁹ is C(18, 9). This is the coefficient of x⁹ in (1+x)⁹(1+x)⁹.

   Σ_(K=1)¹⁰ K²C(10, K)² = 100Σⱼ₌₀⁹C(9, j)² = 100C(18, 9) = 100 × 48620 = 4862000

3. So. 22000 L = 4862000, which gives L = 4862000/22000 = 221.

The idea to keep. When a k multiplies C(n, k), absorb it with kC(n, k) = nC(n-1, k-1). A sum of squared coefficients is then Σ C(m, j)² = C(2m, m).

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