The number of 5 letter words formed with the letters of the word CALCULUS is divisible by :

Options

A. 2
B. 3
C. 5
D. 7

Answer

To solve the problem of finding the number of 5-letter words that can be formed using the letters of the word "CALCULUS", we need to consider the frequency of each letter and the different combinations and arrangements that can be made.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies**:
The word "CALCULUS" consists of the following letters:
- C: 2
- A: 1
- L: 2
- U: 2
- S: 1

2. **Total Letters Available**:
We have a total of 8 letters (C, A, L, C, U, L, U, S).

3. **Different Cases for Forming 5-Letter Words**:
We can form 5-letter words in different ways based on the repetition of letters. The possible cases are:
- Case 1: 5 different letters
- Case 2: 2 letters are the same, and 3 are different
- Case 3: 2 pairs of letters are the same, and 1 letter is different

4. **Case 1: 5 Different Letters**:
The letters we can use are C, A, L, U, S.
- Total letters = 5 (C, A, L, U, S).
- The number of arrangements = 5! = 120.

5. **Case 2: 2 Same Letters and 3 Different Letters**:
We can choose one pair from C, L, or U. The remaining letters must be selected from the other letters.
- Choose 2 same letters (C, L, or U): 3 choices.
- Choose 3 different letters from the remaining letters.
- For example, if we choose CC, we can select from A, L, U, S (4 letters).
- The number of ways to choose 3 letters from 4 = C(4, 3) = 4.
- The arrangement for each selection = 5! / 2! = 60.
- Total for this case = 3 * 4 * 60 = 720.

6. **Case 3: 2 Pairs of Same Letters and 1 Different Letter**:
We can choose two pairs from C, L, U and one different letter from the remaining.
- Choose 2 pairs from (C, L, U): C(3, 2) = 3 ways.
- Choose 1 different letter from the remaining letters (A, S): 2 choices.
- The arrangement for each selection = 5! / (2! * 2!) = 30.
- Total for this case = 3 * 2 * 30 = 180.

7. **Total Number of 5-Letter Words**:
Now, we sum the total from all cases:
- Case 1: 120
- Case 2: 720
- Case 3: 180
- Total = 120 + 720 + 180 = 1020.

8. **Divisibility**:
The final step is to determine by which numbers the total (1020) is divisible. The prime factorization of 1020 is:
- 1020 = 2^2 * 3 * 5 * 17.
- Therefore, 1020 is divisible by 1, 2, 3, 4, 5, 6, 10, 12, 15, 17, 20, 30, 34, 51, 60, 68, 85, 102, 170, 204, 340, 510, and 1020.