Let a polynomial function f(x) be defined as f(x) = _ r=0 ¹⁸ x^r r!(18-r)! . If f(1) + f'(1) = A 2^B C! , where A is an odd integer and B, C N , then the value of A + B + C is equal to

Step-by-step solution

f(x) = _ r=0 ¹⁸ x^r r!(18-r)! Multiply and divide by 18! : f(x) = 1 18! _ r=0 ¹⁸ 18! r!(18-r)! x^r = 1 18! _ r=0 ¹⁸ ¹⁸C_ r x^r f(x) = (1+x)¹⁸ 18! Substitute x=1 : f(1) = 2¹⁸ 18! Differentiate f(x) with respect to x : f'(x) = 18(1+x)¹⁷ 18! = (1+x)¹⁷ 17! Substitute x=1 : f'(1) = 2¹⁷ 17! Now, add f(1) and f'(1) : f(1) + f'(1) = 2¹⁸ 18! + 2¹⁷ 17! = 2¹⁸ + 18 2¹⁷ 18! = 2¹⁷(2 + 18) 18! = 20 2¹⁷ 18! = 5 2¹⁹ 18! Comparing this with A 2^B C! , we get A = 5 (which is an odd integer), B = 19 , and C = 18 . Therefore, A + B + C