2007 UNCO Math Contest II Problems/Problem 10

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Problem

A quaternary “number” is an arrangement of digits, each of which is $0, 1, 2, 3.$ Some examples: $001, 3220, 022113.$

(a) How many $6$-digit quaternary numbers are there in which each of $0, 1$ appear at least once?

(b) How many $n$-digit quaternary numbers are there in which each of $0, 1, 2,$ appear at least once? Test your answer with $n=3.$

(c) Generalize.


Solution

(a) $4^6-2\cdot3^6+2^6$

(b) $4^n-3\cdot 3^n+3\cdot 2^n-1^n$

(c) Generalize

See Also

2007 UNCO Math Contest II (ProblemsAnswer KeyResources)
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