Q. 1.6 - A First Course in Probability

Question

How many different $7$-place license plates are possible when $3$ of the entries are letters and $4$ are digits? Assume that repetition of letters and numbers is allowed and that there is no restriction on where the letters or numbers can be placed.

Step-by-Step Solution

Answer

The possible no. of license plates are $35\times {\left(26\right)}^3\times {\left(10\right)}^4$.

Step 1. Given information.

Total no. of letters and digits on a license plate $=7$

No. of letters on a license plate $=3$

No. of digits on a license plate $=4$

Digits and letters can be repeated and can be placed at any place.

Step 2. Find the possible no. of license plates.

No. of ways of selecting $3$ letters out of $7$ places $=\frac{7!}{3!4!}=\frac{7\times 6\times 5\times 4!}{3\times 2\times 1\times 4!}=35$

Out of $26$ letters, $3$ letters can be selected (with repetition) in $26!\times 26!\times 26! \text{ways} = {\left(26\right)}^3$

Out of $10$ digits, $4$ digits can be selected (with repetition) in $=10!\times 10!\times 10!\times 10! \text{ways} = {\left(10\right)}^4$

Therefore, the possible no. of license plates are $35\times {\left(26\right)}^3\times {\left(10\right)}^4$.