(180)

Given,

R = {a, 6, c, d, e} and S = {1, 2,3,4}

Now taking, f(a) = 1 we get, one of f(b), f(c), f(d), f(e) = 1 then total such cases = 4.3! = 24

Now if, only f(a) = 1, then we have distribute {2,3,4} amongst {b, c, d, e},

So, total cases = 3 4 - = 36

So, number of onto functions when f(a) = 1 is 24 + 36 = 60

Now finding, total number of onto functions,

= 4 5 -( 4 C l .3 5 ) + ( 4 C 2 .2 5 )-( 4 C 3 .l)

= 1024 - 973 + 192-4

= 240

Number of required functions when f(a) 1 will be,

= 240 - 60= 180

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