In how many ways we can select 4 letters from the letters of the word MISSISSIPPI ?

Answer

To solve the problem of selecting 4 letters from the word "MISSISSIPPI," we first need to analyze the frequency of each letter in the word.

1. **Count the frequency of each letter:**
- M: 1
- I: 4
- S: 4
- P: 2

2. **Identify the cases for selecting 4 letters:**
We can categorize the selection based on the number of identical letters:
- Case 1: All four letters are identical.
- Case 2: Three letters are identical, and one is different.
- Case 3: Two letters are identical, and the other two are different.
- Case 4: Two pairs of identical letters.
- Case 5: All letters are different.

3. **Calculate the number of ways for each case:**

- **Case 1: All four are identical.**
- The only letters that can be chosen four times are I and S.
- Therefore, the number of ways = \( \binom{2}{1} = 2 \) (choosing either I or S).

- **Case 2: Three are identical and one is different.**
- We can have three I's or three S's.
- If we choose three I's, we can choose one from {M, S, P} (3 options).
- If we choose three S's, we can choose one from {M, I, P} (3 options).
- Therefore, the number of ways = \( \binom{2}{1} \times 3 = 2 \times 3 = 6 \).

- **Case 3: Two are identical and the other two are different.**
- We can have two I's or two S's, and we need to choose two different letters from the remaining.
- If we choose two I's, we can choose two from {M, S, P} (3 letters), which gives \( \binom{3}{2} = 3 \).
- If we choose two S's, we can again choose two from {M, I, P} (3 letters), which gives \( \binom{3}{2} = 3 \).
- Therefore, the number of ways = \( \binom{2}{1} \times 3 = 2 \times 3 = 6 \).

- **Case 4: Two pairs of identical letters.**
- We can select two I's and two S's or two I's and two P's or two S's and two P's.
- The only valid combinations are (2 I's, 2 S's) and (2 I's, 2 P's) and (2 S's, 2 P's).
- Therefore, the number of ways = \( \binom{3}{2} = 3 \).

- **Case 5: All letters are different.**
- We can select one from each of the letters M, I, S, and P.
- Therefore, the number of ways = 1.

4. **Summing up all the cases:**
- Total ways = Case 1 + Case 2 + Case 3 + Case 4 + Case 5
- Total ways = 2 + 6 + 6 + 3 + 1 = 18.

Thus, the total number of ways to select 4 letters from the word "MISSISSIPPI" is **18**.