Question: A computer outputs a hexadecimal code consisting of a string x1x2x3x4x5 of 5 symbols, each of which is one of {0,1,...,9,A,B,C,D,E,F}. Assume that the output

A computer outputs a hexadecimal code consisting of a string x1x2x3x4x5

of 5 symbols, each of which is one of {0,1,...,9,A,B,C,D,E,F}. Assume that the output is

chosen uniformly at random from among all codes of this form. Let A be the event

that |{x1,x3,x4}|= 1 and B be the event that |{x1,x3,x4}|= 3.

(i) What is Pr(A)?

(ii) What is Pr(B)?

Consider the set F of all functions f : {A,B,C} {1,2,3,4,5}. Let SAB be the subset of F consisting of functions f for which f(A) = f(B). Similarly, let SAC be the subset of F consisting of functions f for which f(A) = f(C) andlet SBC be the subset of F consisting of functions f for which f(B) = f(C).

(i) Find |F|.

(ii) Find |SAB|.

(iii) Use the principle of inclusion-exclusion to work out how many functions

in F are one-to-one. You may assume that |SAB| = |SAC| = |SBC|.

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