Let A = {a, b, c, d}.

a. Calculate the number of subsets of A.
b. Calculate the number of permutations of A.
c. Calculate the number of permutations of A which have no fixed points.
d. Calculate the number of functions from A to A

a. 16

b. 24

c.  9

d. 256

Step-by-step explanation:

Let cardinality of the set A .

Any subset of A can contain i elements.

Now these i elements can be chosen in ways. So the number of subsets can be written as,

a. Here we have n = 4. So the Total no. of subsets = .

b. The no. of permutations is n! for any set with cardinality n. So, here it is = 4! = 24

c. Let denote the set consisting of all permutations of A where i is fixed, . Using symmetry, (fix one element and permute the rest) is the same . Also (fix 2 elements and permute the rest).

By similar arguments, and .

Recall the Principle of inclusion exclusion,

Note that is the set containing permutations with at least one fixed point. So we require 4! - S.

Computing S.

Required answer is 4! - S = 24 - 15 = 9

d. In general the no.  of functions from A (|A| = n) to B (|B| = m) is given by, . Any element of A can be assigned to any of the m elements in B, so the possibilities are .

Here m = n = 4. So the answer is .