Question

For each word below, determine the number of rearrangements of the word in which all letters must be used.

a. OVERNUMEROUSNESSES

Answer 1

To determine the number of rearrangements of the word 'OVERNUMEROUSNESSES', use the formula for permutations with repetition.

Step-by-step explanation:

To determine the number of rearrangements of the word 'OVERNUMEROUSNESSES', we need to find the permutation of all the letters in the word. The word has a total of 17 letters, but some letters are repeated. We can calculate the number of rearrangements using the formula for permutations with repetition.

Step 1: Determine the total number of letters in the word: 17

Step 2: Determine the frequency of each letter in the word:

1. O: 1
2. V: 1
3. E: 5
4. R: 1
5. N: 1
6. U: 2
7. M: 1
8. S: 4

Step 3: Calculate the number of rearrangements using the formula for permutations with repetition:
N! / (n1! * n2! * ... * nk!), where N is the total number of letters and n1, n2, ..., nk are the frequencies of each letter.

Using this formula, we get: 17! / (1! * 1! * 5! * 1! * 1! * 2! * 1! * 4!) = 101,606,736,352,000

Therefore, there are 101,606,736,352,000 possible rearrangements of the word 'OVERNUMEROUSNESSES' in which all letters must be used.