In how many ways can the letters of the word FACTORING be arranged, without changing the order of the vowels a, o, i
To solve this problem, first, we need to identify the total number of arrangements of the letters in the word FACTORING. Then we need to subtract the number of arrangements that would result from changing the order of the vowels 'a', 'o', and 'i'.
Step 1: Total number of arrangements
The word FACTORING has a total of 9 letters. To find the number of arrangements, we use the formula for permutations of a word with repeated letters.
In this case, we have:
- 2 'F's
- 1 'A'
- 1 'C'
- 1 'T'
- 1 'O'
- 1 'R'
- 1 'I'
- 1 'N'
- 1 'G'
The formula for permutations with repeated letters is:
Number of arrangements = (Total number of letters)! / ((Number of repetitions for each letter)!)
So, the total number of arrangements of the letters in the word FACTORING is:
9! / (2! * 1! * 1! * 1! * 1! * 1! * 1! * 1! * 1!)
Step 2: Number of arrangements with vowels in fixed order
To calculate the number of arrangements with the vowels 'a', 'o', and 'i' in a fixed order, we treat them as a single letter.
So, the vowels 'a', 'o', and 'i' become 'AOI'. Now we have 7 letters to arrange: 'F', 'C', 'T', 'R', 'N', 'G', and 'AOI'.
Using the same formula as above, the number of arrangements of these 7 letters is:
7! / (1! * 1! * 1! * 1! * 1! * 1! * 1!)
Step 3: Subtracting the number of arrangements with fixed vowels
To get the final answer, we subtract the number of arrangements with fixed vowels from the total number of arrangements.
Final answer = (Total number of arrangements) - (Number of arrangements with vowels in fixed order)
Substituting the values, we get:
Final answer = (9! / (2! * 1! * 1! * 1! * 1! * 1! * 1! * 1! * 1!)) - (7! / (1! * 1! * 1! * 1! * 1! * 1! * 1!))
Now you can calculate both values and subtract to find the final answer.