# Combinations with Restrictions

Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly one letter is repeated exactly once, but digits cannot be repeated?

We have $$\binom{4}{2}$$ = 6 positions for the two identical letters to occupy

And we have (25)(24) ways to choose the other two letters The total number of "words" = 6 * 26 *25*24 = 93600

Since the digits cannot be repeated = 10 * 9 = 90

The total possibilities = 93600 *90 =8,424,000

Is this correct?

• I think it is right – Tojrah Apr 18 at 17:50

Let's say the repeated letter is $$A$$; you are correct that then number of ways we can put two $$A$$ on a plate is $$6$$. The are $$25 \cdot 24$$ ways to pick the other two letters. But what if repeated letter is $$B$$? We get the same number of arrangements. So I think the correct number of arranging letters is $$26 \cdot 25 \cdot 24 \cdot 6$$. Everything looks correct.