In how many ways we can select 3 letters of the word PROPORTION?

Answer

To solve the problem of selecting 3 letters from the word "PROPORTION," we will break it down into different cases based on the repetition of letters. The letters in "PROPORTION" are as follows:

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

### Step 1: Identify the Cases
We will consider three cases based on the selection of letters:
1. All three letters are different.
2. Two letters are the same, and one letter is different.
3. All three letters are the same.

### Step 2: Case 1 - All Three Letters are Different
We first need to identify the unique letters in "PROPORTION":
- Unique letters: P, R, O, T, I, N (total of 6 unique letters)

To select 3 different letters from these 6 unique letters, we use the combination formula:
\[ \text{Number of ways} = \binom{6}{3} \]
Calculating this gives:
\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 3: Case 2 - Two Letters are the Same, One is Different
In this case, we can have pairs of letters that are the same:
- Possible pairs: PP, RR, OO

We can select one of these pairs and then choose one different letter from the remaining letters.

1. If we select PP, the remaining letters are R, O, T, I, N (5 options).
2. If we select RR, the remaining letters are P, O, T, I, N (5 options).
3. If we select OO, the remaining letters are P, R, T, I, N (5 options).

Thus, for each pair, we have 5 choices for the different letter. Since there are 3 pairs, the total number of ways is:
\[ 3 \times 5 = 15 \]

### Step 4: Case 3 - All Letters are the Same
The only letter that appears three times is O. Thus, there is only one way to select three O's:
\[ \text{Number of ways} = 1 \]

### Step 5: Total Number of Ways
Now we sum up the number of ways from all three cases:
\[ \text{Total ways} = \text{Case 1} + \text{Case 2} + \text{Case 3} = 20 + 15 + 1 = 36 \]

### Final Answer
The total number of ways to select 3 letters from the word "PROPORTION" is **36**.

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