The value of the sum _ k=3 ¹⁰ , ¹⁰C_ k ( 3^ k-2 + 2^ k-2 ) is equal to:

Step-by-step solution

We are given the sum S = _ k=3 ¹⁰ ¹⁰C_ k ( 3^ k-2 + 2^ k-2 ) . This can be split into two sums: S = _ k=3 ¹⁰ ¹⁰C_ k 3^ k-2 + _ k=3 ¹⁰ ¹⁰C_ k 2^ k-2 Using the binomial expansion, we know that: (1+x)¹⁰ = ¹⁰C₀ + ¹⁰C₁ x + _ k=2 ¹⁰ ¹⁰C_ k x^k Rearranging and dividing by x^2 , we get: _ k=2 ¹⁰ ¹⁰C_ k x^ k-2 = (1+x)¹⁰ - 1 - 10x x^2 Since our sum starts from k=3 , we must subtract the k=2 term, which is ¹⁰C₂ x^0 = 45 . For x=3 : _ k=3 ¹⁰ ¹⁰C_ k 3^ k-2 = 4¹⁰ - 1 - 30 9 - 45 = 4¹⁰ - 31 9 - 45 For x=2 : _ k=3 ¹⁰ ¹⁰C_ k 2^ k-2