Let and be finite sets with ( )

MATHEMATICS · Multi-part · CBSE · Mathematics · Permutations and Combinations · Permutations and Combinations

Problem

Let

A

and

B

be finite sets with

|A|=|B|=n

(

n\ge 3

). Fix a subset

S\subset B

with

|S|=k

where

0\le k\le n

. (i) Derive a formula for the number of bijective functions

g:A\to B

in which exactly

k

elements of

A

have their images lying in the specified subset

S

. Express your answer in terms of

n

and

k

. (ii) Using your result, calculate the number of such bijective functions when

n=5

and

k=2