Question

In Iowa, the format for license plates is 2 numbers followed by 4 letters. However the letters I, O, W, A are not allowed and also the use of repeats is not allowed either. How many license plates are available?

Answer 1

32291999

Explanation:

Step 1

We will use the fundamental counting principle to work out the total number of possible plates. The fundamental principle of counting states that if a process can be completed in k steps, and that there are
`n_1` steps to complete the first step,
`n_2` steps to complete the second step and
`n_k` stepts to finish the kth step, then there are
`n_1* n_2 * ....* n_k` ways to complete the steps.

Step 2

In this step we figure out the number of ways to complete each step given that no repetitions are allowed. Note that this will include the plate that has I,O,W,A.

The first step can be carried out in 10 ways because there are 10 digits.

The second step can be carried out in 9 ways because we are not allowed to repeat any digits.

The third step can be carried out in 26 ways as there are 26 letters of the alphabet.

The fourth step can be carried out in 25 ways because no repeats are allowed.

the fifth step can be carried out in 24 ways because two of the letters will already have been used.

The sixth step can be carried out in in 23 ways because three letters have been used already.

Step 3

The third step is to use the fundamental counting principle with the values listed in step 3. We will also subtract 1 from whatever value we get because we would have counted the combination with I,O,W,A in it. This calculation is shown below,

`10* 9* 26* 25* 24* 23-1=32291999`