How many one-to-one functions are there from a set of cardinality 8 to sets with

Question

How many one-to-one functions are there from a set of cardinality 8 to sets with the following cardinalities?

(a) 6, (b) 8, (c) 10, (d) 12.

Please don't just give me the answer give me an explination. Thanks!

Explanation / Answer

PS: Please rate my answer The generic question is - how many one-to-one functions are there from a set with m elements to a set with n elements? (n>m) Lets assume the domain is {x_1,x_2,...,x_m}. Now consider x_1, it can choose one of the n elements in the codomain to be its image. Now consider x_2, it can choose one of the remaining n-1 elements in the codomain to be its image. Similarly x_3 can choose one of the remaining n-2 elements in the codomain to be its image and so on until x_m -1 And x_m can choose one of the remaining n - (m-1) elements in the codomain to be its image. So the number of such functions is n*(n-1)*(n-2)*...*(n-(m-1)). a. m=8, n=6 here m>n, this is not one-to-one. There is a corollary saying - "A function from a set with m elements to a set with n elements is not one-to-one if m > n." b. m=8, n=8 8*7*...*1 = 8! c. m=8, n=10 10*9*....*3 d. m=8, n=12 12*11*....*5

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