how do we calculate the number of one one. funtions from one set with m elements to a set with n elements..please help...also if a set A has 4 elements and Set B has 4 elements what is the probbaility of getting a bijection if a function is drawn from A to B

Dear Rahul,

ts say A = {a,b,c,d} and B = {s,t,u}. we want to find the number of onto functions f:A→B.

well, some member of B has 2 pre-images. there are 3 ways (= n) this can happen. having made THAT choice, we have to count the number of ways we can pick two elements of A to go to that member of B. so we need to know how many ways there are to pick 2 things out of 4 things, that number is "4 choose 2" or:

4!/(2!(4-2)!) = 4!/(2!2!) = 24/((2)(2)) = 24/4 = 6 (this is the "stuff inside the parentheses", which is called a binomial coefficient).

having decided which 2 elements of A get "doubled up" onto which element of B, we now have to assign the remaining targets for the rest of A.

we have 2 choices for where to send the 3rd element of A (since one element of B is already spoken for), after which our last choice is already made for us.

so we have (2)(1) = 2! ways to assign the rest of A, for a grand total of 3(6)(2) = 36 possible onto functions.

Cracking IIT just got more exciting,It s not just all about getting assistance from IITians, alongside Target Achievement and Rewards play an important role. ASKIITIANS has it all for you, wherein you get assistance only from IITians for your preparation and win by answering queries in the discussion forums. Reward points 5 + 15 for all those who upload their pic and download the ASKIITIANS Toolbar, just a simple  to download the toolbar….

So start the brain storming…. become a leader with Elite Expert League ASKIITIANS

Thanks

Aman Bansal

Askiitian Expert