Question

There are two sets S and T. |S|=4 and |T|=10, how many one to one functions and onto functions can be made from these sets?

Answer #1

For one to one functions, each element of S will be mapped to only one distinct element of T. That is we need to select 4 elements out of 10 elements of T where ordering of selection is important.

Number of one to one functions = ^{10}P_{4} =
10! / (10-4)! = 10! / 6!

= 10 * 9 * 8 * 7

= 5040

For onto functions, each element of S is mapped to at least one element of T.

Total number of ways in which 4 elements of T can be uniquely
mapped to 4 elements of S = ^{4}P_{4 }=
24

Total number of ways in which rest 6 elements can be mapped to
any of 4 elements of S = 4^{6}

Total number of onto functions = 24 * 4^{6} = 98304

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