In how many ways can the letters of the word OBASANJO be arranged such that the vowels will not be together
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.
To find the arrangements where the vowels are not together in the word "OBASANJO," we can use the principle of inclusion-exclusion.
Step 1: Find the total number of arrangements without any restrictions.
The word "OBASANJO" has a total of 8 letters, but there are repeated letters: 'O' appears twice and 'A' appears twice.
Therefore, the total number of arrangements without any restrictions is 8!/2!2! = 20160.
Step 2: Find the number of arrangements where the vowels are together.
To calculate this, consider the vowels as a single unit, 'OAAO'. Now, there are 5 units, including the grouped vowels and the other consonants ('B', 'S', 'N', 'J'). The number of arrangements of these 5 units is 5!.
However, within the grouped vowels, 'A' is repeated twice. So, we need to divide by 2! to eliminate the overcounting. Therefore, the number of arrangements where the vowels are together is 5! / 2! = 60.
Step 3: Find the number of arrangements where the vowels are not together.
To find the number of arrangements where the vowels are not together, subtract the number of arrangements found in Step 2 from the total number of arrangements found in Step 1.
Number of arrangements where the vowels are not together = Total number of arrangements - Number of arrangements where the vowels are together
= 20160 - 60
= 20100.
Therefore, there are 20,100 ways to arrange the letters of the word "OBASANJO" such that the vowels are not together.