The number of ways in which an arrangement of 4 letters of the word proportion can be made is

Options

A. 700
B. 750
C. 758
D. 800

Answer

To find the number of ways to arrange 4 letters from the word "proportion," we need to consider the frequency of each letter in the word. The letters in "proportion" are as follows:

- P: 2 times
- R: 2 times
- O: 3 times
- I: 1 time
- T: 1 time
- N: 1 time

Now, we will analyze different cases based on the repetition of letters.

### Step 1: Case 1 - All four letters are distinct
We can choose 4 distinct letters from the available letters (P, R, O, I, T, N). The total number of distinct letters is 6 (P, R, O, I, T, N).

To find the number of ways to choose 4 letters from these 6, we use the combination formula:

\[ \text{Number of ways to choose 4 letters} = \binom{6}{4} \]

After choosing the letters, we can arrange them in \(4!\) ways.

Calculating this:

\[ \binom{6}{4} = \frac{6!}{4! \cdot (6-4)!} = \frac{6 \times 5}{2 \times 1} = 15 \]

Thus, the total arrangements for this case is:

\[ 15 \times 4! = 15 \times 24 = 360 \]

### Step 2: Case 2 - Three letters are the same, one is different
In this case, we can only have 'O' as the letter that appears three times (since it appears 3 times in "proportion"). We can choose one more letter from the remaining letters (P, R, I, T, N), which gives us 5 options.

The arrangement for this case is:

\[ \text{Number of arrangements} = \frac{4!}{3! \cdot 1!} \cdot 5 = 4 \cdot 5 = 20 \]

### Step 3: Case 3 - Two letters are the same, and two are different
We can have two P's or two R's.

1. **Two P's**: We can choose 2 more letters from (R, O, I, T, N) which gives us 5 options. The number of ways to choose 2 letters from 5 is \(\binom{5}{2}\).

The arrangements are:

\[ \text{Number of arrangements} = \frac{4!}{2! \cdot 2!} \cdot \binom{5}{2} = 6 \cdot 10 = 60 \]

2. **Two R's**: Similarly, we can choose 2 more letters from (P, O, I, T, N) which also gives us 5 options.

The arrangements are the same as above:

\[ \text{Number of arrangements} = 6 \cdot 10 = 60 \]

So, the total for this case is:

\[ 60 + 60 = 120 \]

### Step 4: Case 4 - Two letters are the same, and two are different (with O)
We can take two O's and choose 2 letters from (P, R, I, T, N). The number of ways to choose 2 letters from 5 is \(\binom{5}{2}\).

The arrangements are:

\[ \text{Number of arrangements} = \frac{4!}{2! \cdot 2!} \cdot \binom{5}{2} = 6 \cdot 10 = 60 \]

### Step 5: Total arrangements
Now, we add all the cases together:

- Case 1: 360
- Case 2: 20
- Case 3: 120
- Case 4: 60

Total arrangements:

\[ 360 + 20 + 120 + 60 = 560 \]

### Final Answer
The total number of ways to arrange 4 letters from the word "proportion" is **560**.