Count the number of functions phi 4 58 rightarrow 4 58 so th

Count the number of functions phi: (4, 5....8) rightarrow (4, 5......8) so that Phi is one-to-one and phi (5) = 5 Phi is one-to-one and Phi(5) notequalto 5

Solution

i)

5 entries in domain and 5 in codomain

Function is one to one . phi(5) is fixed

So only other 4 need to be fixed

Function is one to one so no two element have the same image

4 can map to 4 other elements

6 can map to 3 othr elements

7 to 2 othr elements

8 to last remaining elements

So, 4*3*2=24 maps

b)

Since there is restriction on 5 so we first map 5 to any of the other 4 elements

4 can be mapped to 4 remaining elements

6 to 3 remaining elements

7 to 2 remaining elements

8 to the last remaining elements

SO, 4*4*3*2=96 maps