Ukrainian National Mathematical Olympiad

Question

Find the exact value of the expression below: (6!+5!)(5!+4!)(4!+3!)(3!+2!)(2!+1!)(6!-5!)(5!-4!)(4!-3!)(3!-2!)(2!-1)! if n! denotes the product 1 2 3 n for every natural number n.

Accepted Answer

Using the obvious equality (n+1)! = (n+1) n! we can make the following transformations: (6!+5!)(5!+4!)(4!+3!)(3!+2!)(2!+1!)(6!-5!)(5!-4!)(4!-3!)(3!-2!)(2!-1)! = 5!(6+1) 4! (5+1) 3! (4+1) 2! (3+1) 1! (2+1)!5!(6-1) 4! (5-1) 3! (4-1) 2! (3-1) 1! (2-1)! = 7 6 5 4 35 4 3 2 1 = 7 62 1 = 21.