Question

In how many ways can the letters of the word OBASANJO be arranged such that the vowels will not be together

Answer 1

First, we can count the total number of arrangements of the letters in OBASANJO without any restrictions.

There are 8 letters, so there are 8 choices for the first letter, 7 for the second, 6 for the third, and so on, giving a total of:

8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320

Now, let's count the number of arrangements where the vowels are together. If we treat the three vowels (O, A, O) as a single unit, then there are 6 units to arrange:

- 1 unit for the block of vowels
- 1 unit for the remaining O
- 1 unit for the remaining B
- 1 unit for the remaining S
- 1 unit for the remaining A
- 1 unit for the remaining N
- 1 unit for the remaining J

There are 6! ways to arrange these units, and within the block of vowels, there are 3! additional ways to arrange the vowels themselves. So the total number of arrangements where the vowels are together is:

6! x 3! = 2,160

To get the number of arrangements where the vowels are not together, we can subtract the number of arrangements where the vowels are together from the total number of arrangements:

40,320 - 2,160 = 38,160

Therefore, there are 38,160 ways to arrange the letters of the word OBASANJO such that the vowels will not be together.