Find the number of elements in the set sigma epsilon S5 sigm

Find the number of elements in the set {sigma epsilon S_5: sigma (2) = 5}.

Solution

S₅ is the symmetric group on 1,2,3,4,5.

1 has 4 options( 1 can be mapped to 1 or 2 or 3 or 4), 2 has single option (since 2 is fixed with 5), 3 has 3 options,4 has 2 options, 5 has single option.Hence the required number of elements in S5 is 4.3.2.1=24.