Let A and B be finite sets and P A and P B , respectively denote their power sets. If P B has 112 elements more than those in P A , then the number of functions from A to B which are injective is

Step-by-step solution

Consider N A = m   and   N B = n P B has 112 elements more than P A N P B = 112 + N P A ⇒ N P B - N P A = 112 ⇒ 2 n - 2 m = 16 × 7 Rewrite the above equation. 2 m 2 n - m - 1 = 16 × 7 ⇒ 2 m 2 n - m - 1 = 2 4 2 3 - 1 From the above equation m = 4 ⇒ n - m = 3 ⇒ n - 4 = 3 ⇒ n = 7 The number of injective function from A to B = P N ( A ) N ( B ) = P 4 7 = 7 ! 3 ! = 840