Plates with letters and distinct digits

Text

A licence plate consists of $2$ letters (chosen from a $21$-letter alphabet, repetitions allowed) followed by $3$ digits from $0$ to $9$ that are all different. How many distinct plates can be formed?

Solution

We apply the product rule, counting the two parts separately.

• The $2$ letters allow repetition: $21 \cdot 21 = 21^2 = 441$ ways.
• The $3$ digits must be distinct: these are simple arrangements, $10 \cdot 9 \cdot 8 = 720$ ways.

In total: $N = 441 \cdot 720 = \boxed{317\,520}.$