Discussion

Let S be the set of all 5-digit numbers having only the digits 0 and 1. Then n (S) =

Answer: A

\(n(S) =\) 16.

A 5-digit number cannot begin with 0, so the first digit must be 1. Each of the other four places can be 0 or 1:

\[ n(S) = 1 \times 2^{4} = 16 \]

32 counts all \(2^{5}\) strings, including those starting with 0, which are not 5-digit numbers. 8, 24 and 64 do not come from this count.

The same reasoning gives \(9 \times 10^{4}\) for all 5-digit numbers with digits 0 to 9: the leading digit has one fewer choice than the others.

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