Question

How many times can the letters of the word carrier be arranged if the first letter is r and the next letter is not r?

Answer 1

240 ways

Explanation:

We have the word CARRIER and we're looking for the number of ways we can arrange the letters, given that we want the first letter to be R and the next letter to not be R.

Let's set an R as the first letter. That leaves CARRIE to work with.

The next letter is not an R. There are 4 unique letters that are not R: C, A, I, E. We have 4 choices. For simplicity, let's just set the E as the next letter. We're left with CARRI.

Now we have 5 letters left to arrange in 5 places. However - there are still two R's in the bunch, so we need to eliminate the duplicates.

If the remaining 5 letters were all unique, we'd have `5!` ways to arrange the 5 letters. However, they are not. To get rid of the duplicates, we divide by the number of ways we can internally arrange the 2 Rs (which is `2!`).

All in all, we have:

`4xx(5!)/(2!)=4xx120/2=4xx60=240` ways to arrange the letters