Find the number of surjective functions

[Moon]

Let $m \geq n > 0$. Find the number of surjective functions from $B_m =\{1, 2, \cdots , m\}$ to $B_n = \{1,2,\cdots , n\}$

[sourav das]

Bhaia, I don't know whether my ans. is right or wrong;But i use exclusion-inclusion to solve it.

And the ans. is:

\[n^{m}-\binom{n}{1}\left ( n-1 \right )^{m}+\binom{n}{2}(n-2)^{m}-...............+(-1)^{n-1}n\]

Please reply.

[Moon]

Please give complete solution. Your answer seems to be correct, though!

[sourav das]

There are \[n^{m}\] ways to make any function.
There are \[\binom{n}{1}\left ( n-1 \right )^{m}\] ways to make a function not using one element from co-domain.
There are \[\binom{n}{2}\left ( n-2 \right )^{m}\]ways to make a function not using two elements from co-domain.
In this way,
.
.
.
There are \[\binom{n}{n-1}\left ( n-n+1 \right )^{m}= n\]ways to make a function not using n-1 elements from co-domain.
Now using exclusion-inclusion method and find the number of surjective function:

\[n^{m}-\binom{n}{1}\left ( n-1 \right )^{m}+\binom{n}{2}(n-2)^{m}-...............+(-1)^{n-1}n\]
Do i have to give more explaination?