Let `A={1,\ 2,\ ,\ n}` and `B={a ,\ b}` . Then the number of subjections from `A` into `B` is `\ ^n P_2` (b) `2^n-2` (c) `2^n-1` (d) `\ ^n C_2`

Options

A. ` ""^(n)P_(2)`
B. `2^(n)-2`
C. `2^(n)-1`
D. None of these

Answer

D

Explanation

Given that, A = { 1, 2, 3, …, n} and B = {a, b}.
We know that, if A and B are two non-empty finite sets containing m and n elements respectively, then the number of surjection from A into B is
`""^(n)C_(m)xxm!, if n ge m`
0, if `n lt m`
Here, m = 2
` :. ` Number of surjection from A into B is
`""^(n)C_(2)xx2! =(n!)/(2!(n-2)!)xx2!`
` " " =(n(n-1)(n-2)!)/(2xx1(n-2))xx2! =n^(2)-n`