how many bijection function are possible if domain has m elements,codomain has n elements

for a f(x) to be bijective, m should be equal to n. this is bcause if m>n, ther would be atlest one element in codomain which would have more than one preimage making the function manyone. If m<n, ther would be atleast one element in codomain which has no preimage making function into.

no. of choices for 1st element in domain to have an images= n

no. of choices for 2nd element to have image= n-1  and so on......

hence no. of bijections =n*(n-1)*(n-2)*.....*2*1 =n!.

Bijective function gives an exact pairing of the elements of two sets. Every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements
So M=N and each element of one set should be paired with one and only one element of another