Question

How many license plates consisting of three letters followed by three digits contain no letter or digit twice?

Short answer

The required license plates are\(11,232,000\).

Step-by-step solution

Definitions of Product rule, Permutation, and Combination

Product rule: If one event can occur in\(m\)ways and a second event can occur in\(n\)ways, then the number of ways that the two events can occur in sequence is then\(m \cdot n\).

Definition permutation (order is important):

\(P(n,r) = \frac{{n!}}{{(n - r)!}}\)

Definition combination (order is not important):

\(C(n,r) = \left( {\begin{array}{*{20}{l}}n\\r\end{array}} \right) = \frac{{n!}}{{r!(n - r)!}}\)with\(n! = n \cdot (n - 1) \cdot \ldots \cdot 2 \cdot 1\)

Calculate the number of strings by the formula of Product rule, Permutation or Combination

The order of the letters/digits is important (since different order leads to different license plates). So, use the formula of permutation.

Select\(3\)letters from the\(26\)letters in the alphabet.

\(\begin{array}{l}P(26,3) = \frac{{26!}}{{(26 - 3)!}}\\P(26,3) = \frac{{26!}}{{23!}}\\P(26,3) = 15,600\end{array}\)

Select\(3\)digits from the\(10\)possible digits \((0,1,2,3,4,5,6,7,8,9)\).

\(\begin{array}{l}P(26,3) = \frac{{10!}}{{(10 - 3)!}}\\P(26,3) = \frac{{10!}}{{7!}}\\P(26,3) = 720\end{array}\)

Use the product rule:

\(15,600 \cdot 720 = 11,232,000\)

Hence, the required license plates are\(11,232,000\).