Chapter 13, Problem 11CT

On February $20$, $2004$, the Ohio Bureau of Motor Vehicles unveiled the state’s new license plate format. The plate consists of three letters $\left(A-Z\right)$ followed by $4$ digits $\left(0-9\right)$. Assume that all letters and digits may be used, except that the third letter cannot be $O,I$ or $Z$. If repetitions are allowed, how many different plates are possible?

To determine

To calculate: The possible number of different license plates, if the plates consist of three letters $\left(A-Z\right)$ followed by 4 digits $\left(0-9\right)$. All digits and numbers may be used, except that the third letter cannot be $O,I$ or $Z$. And repetition is allowed.

Answer to Problem 11CT

Solution:

Total $155,480,000$ license plates are possible according to new license plates format.

Explanation of Solution

Given information:

The plates consist of three letters $\left(A-Z\right)$ followed by 4 digits $\left(0-9\right)$. All digits and numbers may be used, except that the third letter cannot be $O,I$ or $Z$. And repetition is allowed.

Formula used:

Multiplication principle of counting:

If there are $p$ selections for first, $q$ selections for the second choice and so on, the task of making these selections can be done in $p\cdot q\mathrm{...}$ different ways.

Calculation:

The plates consist of three letters $\left(A-Z\right)$ followed by $4$ digits $\left(0-9\right)$ and repetition is allowed.

Thus, first $3$ letters has $26$ distinct options and remaining $4$ digits has $10$ distinct options.

But the third letter cannot be $O,I$ or $Z$, so third letter has $26-3=23$ options.

So here, first letter has $26$, second letter has $26$, third letter has $23$, first digit has $10$, second digit has $10$, third digit has $10$ and 4^th digit has $10$ options.

Using multiplication principle, possible license plates are $26\cdot 26\cdot 23\cdot 10\cdot 10\cdot 10\cdot 10=155,480,000$.

Therefore, total $155,480,000$ license plates are possible according to new license plate format.