Let A be any fixed finite set of 4 or more elements Prove th

Let A be any fixed finite set of 4 or more elements. Prove that the number of subsets A is less than the number of permutations of elements of A.

Solution

for subset, it can be set of 1,2,3 or all 4 elements i.e.

1 element= 4p1=4

similarly, 4p2=6

4P3= 4 and 4P4=1

hence, total subsets= 1+4+4+6=15

but permutation is in how many different ways those elements can be arranged i.e. 4!= 24

hence permutation will always be more than subsets

example: A={1,2,3,4}

subset=[1],[2],[3],[4],[1,2],[1,3],[1,4],[2,3],[2,4],[3,4],1,2,3],[1,2,4],[2,3,4],[1,3,4], [1,2,3,4] total 15

permutation: {1,2,3,4}

{1,2,4,3}

{1,3,4,2}

{1,3,2,4}

{1,4,3,2}

{1,4,2,3}. these are sic combinations when i can 1 as constant and moving other 3. similarly, other 18 can be made . hence totl 24 that is more than 15.