How many words of (i) 3 letters, (ii) 4 letters, (iii) 5 letters can be formed by the letters of the word 'COURTESY'?

Answer

To solve the problem of how many words of 3 letters, 4 letters, and 5 letters can be formed by the letters of the word "COURTESY," we will use the concept of permutations and combinations.

The word "COURTESY" consists of 8 distinct letters: C, O, U, R, T, E, S, Y.

### Step-by-Step Solution:

**(i) Number of 3-letter words:**

1. **Select 3 letters from 8 letters:**
We can choose 3 letters from 8 letters using the combination formula \( \binom{n}{r} \):
\[ \text{Number of ways to choose 3 letters} = \binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} \]

2. **Calculate \( \binom{8}{3} \):**
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

3. **Arrange the 3 letters:**
The number of arrangements of 3 letters is given by \( 3! \):
\[ 3! = 6 \]

4. **Total number of 3-letter words:**
\[ \text{Total} = \binom{8}{3} \times 3! = 56 \times 6 = 336 \]

**(ii) Number of 4-letter words:**

1. **Select 4 letters from 8 letters:**
\[ \text{Number of ways to choose 4 letters} = \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} \]

2. **Calculate \( \binom{8}{4} \):**
\[ \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = \frac{1680}{24} = 70 \]

3. **Arrange the 4 letters:**
\[ 4! = 24 \]

4. **Total number of 4-letter words:**
\[ \text{Total} = \binom{8}{4} \times 4! = 70 \times 24 = 1680 \]

**(iii) Number of 5-letter words:**

1. **Select 5 letters from 8 letters:**
\[ \text{Number of ways to choose 5 letters} = \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} \]

2. **Calculate \( \binom{8}{5} \):**
\[ \binom{8}{5} = \binom{8}{3} = 56 \quad (\text{since } \binom{n}{r} = \binom{n}{n-r}) \]

3. **Arrange the 5 letters:**
\[ 5! = 120 \]

4. **Total number of 5-letter words:**
\[ \text{Total} = \binom{8}{5} \times 5! = 56 \times 120 = 6720 \]

### Final Answers:
- Number of 3-letter words: **336**
- Number of 4-letter words: **1680**
- Number of 5-letter words: **6720**