The number of away of selecting 4 letters taking 2 like and 2 different from the letters of the word PROPORTION is

Options

A. (a)30
B. (b)18
C. (c)36
D. (d)48

Answer

To solve the problem of selecting 4 letters from the word "PROPORTION" such that 2 letters are the same and 2 letters are different, we can follow these steps:

### Step 1: Identify the letters in the word "PROPORTION"
The letters in the word "PROPORTION" are: P, R, O, P, O, R, T, I, O, N.
From this, we can see that:
- P appears 2 times
- R appears 2 times
- O appears 3 times
- T, I, N appear 1 time each

### Step 2: Choose the letter that will be repeated
We have three letters that can be chosen to be the repeated letter: P, R, and O. Therefore, we have 3 choices for the letter that will appear twice.

### Step 3: Choose 2 different letters from the remaining letters
After selecting one letter to repeat, we need to select 2 different letters from the remaining letters.

- If we choose P as the repeated letter, the remaining letters are R, O, T, I, N (5 letters).
- If we choose R as the repeated letter, the remaining letters are P, O, T, I, N (5 letters).
- If we choose O as the repeated letter, the remaining letters are P, R, T, I, N (5 letters).

In all cases, we have 5 remaining letters to choose from.

### Step 4: Calculate the number of ways to choose 2 different letters
The number of ways to choose 2 different letters from 5 letters is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

Here, \( n = 5 \) and \( r = 2 \):
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 5: Calculate the total number of ways
Since we have 3 choices for the repeated letter and 10 ways to choose 2 different letters, the total number of ways is:
\[ 3 \times 10 = 30 \]

### Final Answer
The total number of ways of selecting 4 letters taking 2 like and 2 different from the letters of the word "PROPORTION" is **30**.

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