To solve the problem of finding the number of ways to select and arrange four letters from the word "PROPORTION," we will break it down into two parts: (a) selection of letters and (b) arrangement of letters.
### Step-by-Step Solution:
#### Part (a): Selection of Four Letters
1. **Identify the Letters and Their Frequencies:**
The word "PROPORTION" consists of the following letters:
- P: 2
- R: 2
- O: 3
- T: 1
- I: 1
- N: 1
2. **Case Analysis for Selection of 4 Letters:**
We will consider different cases based on the repetition of letters.
- **Case 1: All letters are different.**
- Unique letters: P, R, O, T, I, N (total = 6)
- We can select 4 letters from these 6 unique letters.
- Number of ways = \( \binom{6}{4} = 15 \)
- **Case 2: Two letters are the same, and two are different.**
- The letters that can be the same: P, R, O (since they appear more than once).
- Choose 1 letter to be repeated (3 choices: P, R, O).
- Choose 2 different letters from the remaining 5 letters.
- Number of ways = \( \binom{3}{1} \times \binom{5}{2} = 3 \times 10 = 30 \)
- **Case 3: Two letters are the same, and the other two letters are also the same.**
- The only possibility is to choose O (3 O's available).
- Choose 2 letters from P, R (2 choices).
- Number of ways = \( \binom{3}{2} = 3 \)
- **Case 4: Three letters are the same, and one is different.**
- The only letter that can be repeated thrice is O.
- Choose 1 letter from the remaining letters (P, R, T, I, N).
- Number of ways = \( \binom{5}{1} = 5 \)
3. **Total Selection Ways:**
- Total ways = Case 1 + Case 2 + Case 3 + Case 4
- Total = \( 15 + 30 + 3 + 5 = 53 \)
#### Part (b): Arrangement of Four Letters
1. **Using the selections from Part (a), we will calculate arrangements for each case.**
- **Case 1: All different letters.**
- Number of arrangements = \( 4! = 24 \)
- Total arrangements = \( 15 \times 24 = 360 \)
- **Case 2: Two letters are the same, and two are different.**
- Arrangement formula = \( \frac{4!}{2!} = 12 \)
- Total arrangements = \( 30 \times 12 = 360 \)
- **Case 3: Two letters are the same, and the other two letters are also the same.**
- Arrangement formula = \( \frac{4!}{2! \times 2!} = 6 \)
- Total arrangements = \( 3 \times 6 = 18 \)
- **Case 4: Three letters are the same, and one is different.**
- Arrangement formula = \( \frac{4!}{3!} = 4 \)
- Total arrangements = \( 5 \times 4 = 20 \)
2. **Total Arrangement Ways:**
- Total arrangements = Case 1 + Case 2 + Case 3 + Case 4
- Total = \( 360 + 360 + 18 + 20 = 758 \)
### Final Answer:
- (a) The number of ways to select four letters = **53**
- (b) The number of ways to arrange four letters = **758**
