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 letters of the word "MISSISSIPPI", we will break it down into cases based on the repetitions of the letters.

### Step-by-Step Solution:

1. **Count the Letters and Their Frequencies**:
- M: 1
- I: 4
- S: 4
- P: 2

2. **Case 1: All Four Letters are Distinct**:
- The distinct letters are M, I, S, and P.
- We can select 4 letters from these 4 distinct letters in \( \binom{4}{4} = 1 \) way.
- The arrangement of these 4 letters can be done in \( 4! = 24 \) ways.
- Total for this case = \( 1 \times 24 = 24 \).

3. **Case 2: Two Letters are Repeated**:
- The repeated letters can be selected from I, S, or P.
- Choose 1 letter to repeat (from I, S, P) and 2 more distinct letters from the remaining letters.
- The combinations are:
- Choose I (2 times), then choose 2 from {M, S, P}: \( \binom{3}{2} = 3 \)
- Choose S (2 times), then choose 2 from {M, I, P}: \( \binom{3}{2} = 3 \)
- Choose P (2 times), then choose 2 from {M, I, S}: \( \binom{3}{2} = 3 \)
- Total combinations = \( 3 + 3 + 3 = 9 \).
- Arrangements for each selection = \( \frac{4!}{2!} = 12 \).
- Total for this case = \( 9 \times 12 = 108 \).

4. **Case 3: One Letter is Repeated Twice and Another Letter is Repeated Twice**:
- The letters that can be repeated are I and S.
- Therefore, we can have combinations of (I, I, S, S).
- Arrangements = \( \frac{4!}{2!2!} = 6 \).
- Total for this case = 6.

5. **Case 4: One Letter is Repeated Thrice and Another Letter Once**:
- Possible letters for repetition are I and S.
- Combinations:
- I (3 times) and choose 1 from {M, S, P}: \( \binom{3}{1} = 3 \)
- S (3 times) and choose 1 from {M, I, P}: \( \binom{3}{1} = 3 \)
- Total combinations = \( 3 + 3 = 6 \).
- Arrangements for each selection = \( \frac{4!}{3!} = 4 \).
- Total for this case = \( 6 \times 4 = 24 \).

6. **Case 5: One Letter is Repeated Four Times**:
- The only letter that can be repeated four times is I.
- Total combinations = 1 (I, I, I, I).
- Arrangements = \( \frac{4!}{4!} = 1 \).
- Total for this case = 1.

7. **Total Combinations**:
- Adding all the cases together:
- Total = \( 24 + 108 + 6 + 24 + 1 = 163 \).

### Final Answer:
The total number of ways to select 4 letters from the letters of the word "MISSISSIPPI" is **176**.