Let K = _ r=2 ⁵⁰ ,⁵⁰C_ r ,^ r C₂ 2^ r-2 . The remainder when K is divided by 11 is

Step-by-step solution

We know that ,^ r C₂ = r(r-1) 2 . Thus, K = 1 2 _ r=2 ⁵⁰ r(r-1) ,⁵⁰C_ r 2^ r-2 . Using the identity r(r-1) ,^ n C_ r = n(n-1) ,^ n-2 C_ r-2 , we have: K = 1 2 _ r=2 ⁵⁰ 50 49 ,⁴⁸C_ r-2 2^ r-2 K = 1225 _ k=0 ⁴⁸ ,⁴⁸C_ k 2^ k K = 1225 (1 + 2)⁴⁸ = 1225 3⁴⁸ . Now, we need the remainder when K is divided by 11 . First, 1225 = 11 111 + 4 1225 4 11 . By Fermat's Little Theorem, 3¹⁰ 1 11 . 3⁴⁸ = (3¹⁰)⁴ 3⁸ 1⁴ 3⁸ 11 . We can evaluate 3⁸ 11 : 3² = 9 -2 11 3⁸ = (-2)⁴ = 16 5 11 . Therefore, K 4 5 = 20 9 11 . The remainder is 9 .