In how many ways can the letters of the word PROPORTION be arranged by taking 4 letters at a time?

Options

A. 123
B. 758
C. 658
D. 578

Answer

To find the number of ways to arrange the letters of the word "PROPORTION" by taking 4 letters at a time, we need to consider the distinct letters and the repetitions of certain letters in the word.

### Step-by-Step Solution:

1. **Identify the Letters and Their Frequencies:**
The word "PROPORTION" consists of the following letters:
- P: 2 times
- R: 2 times
- O: 3 times
- T: 1 time
- I: 1 time
- N: 1 time

Thus, the distinct letters are P, R, O, T, I, N.

2. **Count Distinct Letters:**
There are 6 distinct letters: P, R, O, T, I, N.

3. **Case 1: All 4 Letters are Distinct**
- We can choose 4 letters from the 6 distinct letters.
- The number of ways to choose 4 letters from 6 is given by \( \binom{6}{4} \).
- The arrangement of 4 distinct letters is \( 4! \).

Calculation:
\[ \text{Ways} = \binom{6}{4} \times 4! = 15 \times 24 = 360 \]

4. **Case 2: One Letter Repeats Twice, Two Other Letters are Distinct**
- The letters that can repeat twice are P, R, O.
- We can choose 1 letter from these 3 letters, and then choose 2 more distinct letters from the remaining 5 letters.

Calculation:
- Choose 1 letter to repeat: \( \binom{3}{1} = 3 \)
- Choose 2 distinct letters from the remaining 5: \( \binom{5}{2} = 10 \)
- The arrangement of these letters (2 are the same): \( \frac{4!}{2!} = 12 \)

Total for this case:
\[ \text{Ways} = 3 \times 10 \times 12 = 360 \]

5. **Case 3: Two Letters Repeat Twice**
- We can choose 2 letters from P, R, O.
- The arrangement of these letters (2 of each): \( \frac{4!}{2!2!} = 6 \)

Calculation:
- Choose 2 letters to repeat: \( \binom{3}{2} = 3 \)

Total for this case:
\[ \text{Ways} = 3 \times 6 = 18 \]

6. **Case 4: One Letter Repeats Three Times, One Other Letter is Distinct**
- The only letter that can repeat three times is O.
- We can choose 1 letter from the remaining 5 letters.

Calculation:
- Choose 1 letter from 5: \( \binom{5}{1} = 5 \)
- The arrangement of these letters (3 O's and 1 other): \( \frac{4!}{3!} = 4 \)

Total for this case:
\[ \text{Ways} = 5 \times 4 = 20 \]

7. **Total Arrangements:**
Now, we sum all the cases:
\[ \text{Total} = 360 + 360 + 18 + 20 = 758 \]

### Final Answer:
The total number of ways to arrange the letters of the word "PROPORTION" by taking 4 letters at a time is **758**.