Final answer:
There are 20,160 ways to permute the letters in the word 'research', by calculating 8! (eight-factorial) and dividing by the factorial of the number of occurrences of the letter 'e', which is 2.
Step-by-step explanation:
The number of ways to permute the letters in the word 'research' can be found by calculating the factorial of the number of letters in the word. However, since the letter 'e' is repeated, we have to divide the total by the factorial of the number of times 'e' appears.
The word 'research' has 8 letters with the letter 'e' occurring twice. We use the formula for permutations of a word with repeated letters which is:
n! / (p1! × p2! × ... × pk!)
where n is the total number of letters, and p1, p2, ..., pk are the frequencies of each repeated letter.
For 'research' the calculation would be:
8! / (2! × 1! × ... × 1!) = 40,320 / 2 = 20,160
Therefore, there are 20,160 ways to permute the letters in the word 'research'.