If ^ n +2 C ₈: ^ n -2 P ₄=57: 16 , then ' n ' is

Step-by-step solution

The given ratio is ^ n+2 C₈ ^ n-2 P₄ = 57 16 . Using the definitions ^ n C_ r = n! r!(n-r)! and ^ n P_ r = n! (n-r)! , we have: (n+2)! 8!(n-6)! (n-6)! (n-2)! = 57 16 Simplifying the factorials: (n+2)(n+1)(n)(n-1) 8! = 57 16 (n+2)(n+1)(n)(n-1) 40320 = 57 16 (n+2)(n+1)(n)(n-1) = 57 40320 16 = 57 2520 = 143640 We look for four consecutive integers whose product is 143640 . Testing values for n : If n=19 , the product is 21 20 19 18 = 143640 . Thus, n=19 . Answer: 19