Please send the proof of no:of ontofunctions. How can get the formula.

By using Exclusive & Inclusive formula , which states that

          if A & B are any 2 sets then cardiality of AUB = card (A) + card (B ) – card (A inter B)

          card(AUBUC) = card(A)+card(B)+card(C)-card(A inter B) – card (A inter C)- card (B inter C ) +card (A inter B inter C )

    likewise if we derive for n sets then  it will be

                card (A1 U A2 U A3 …. An ) = ∑_i Ai – ∑_i∑ jAi Aj + ∑i∑j∑k AiAjAk – ….+ (A1 inter A2 ...inter An) , by the help of this we can solve the problem

       let’s take 2 sets having m, n elements & for on to function m>/=n

          if  f : A → B  then total no. of functions possible are n^m but we have to elemenate the functions which do not include one of the “n” elements , so we have to exclude those 

  for that  ⁿC1 (n-1) ^m , so this will become

              n^m – ⁿC₁ (n-1)^ m

  by taking the two of “m” are range,  then  ⁿC₂ (n-2)^m , but now the problem is the terms of

  first term also included , that means we are removing one term twice , so we have to add this , likewise , we will continue upto n-1 elements

     i.e .  n^m – ⁿC₁ (n-1) ^m + ⁿC₂ (n-2 )^m – ….… +(-1) ^(n-1) ^mC_m-1  are the possibilities of on to functions possible from A to B

       this is also written in simple way

                   ⁿ∑_k=0 (-1) ^k ^mC_k (n-k)^m .