Find the total number of perpemutations of the letters of each of the words given below:
(i) APPLE (ii) ARRANGE (iii) COMMERCE
(iv) INSTITUTE (v) ENGINEERING (vi) INTERMEDIATE

Answer

To find the total number of permutations of the letters in each of the given words, we will use the formula for permutations of a multiset:

\[ \text{Number of permutations} = \frac{n!}{p_1! \times p_2! \times \ldots \times p_k!} \]

where \( n \) is the total number of letters, and \( p_1, p_2, \ldots, p_k \) are the frequencies of the repeated letters.

Let's solve each word step by step.

### (i) APPLE

1. **Count the letters**: APPLE has 5 letters.
2. **Identify repetitions**: The letter 'P' repeats 2 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{5!}{2!} = \frac{120}{2} = 60 \]

### (ii) ARRANGE

1. **Count the letters**: ARRANGE has 7 letters.
2. **Identify repetitions**: The letter 'A' repeats 2 times, and 'R' repeats 2 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260 \]

### (iii) COMMERCE

1. **Count the letters**: COMMERCE has 8 letters.
2. **Identify repetitions**: The letter 'C' repeats 2 times, 'M' repeats 2 times, and 'E' repeats 2 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{8!}{2! \times 2! \times 2!} = \frac{40320}{8} = 5040 \]

### (iv) INSTITUTE

1. **Count the letters**: INSTITUTE has 9 letters.
2. **Identify repetitions**: The letter 'I' repeats 2 times, 'T' repeats 2 times, and 'E' repeats 2 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{9!}{2! \times 2! \times 2!} = \frac{362880}{8} = 45360 \]

### (v) ENGINEERING

1. **Count the letters**: ENGINEERING has 11 letters.
2. **Identify repetitions**: The letter 'E' repeats 3 times, 'N' repeats 2 times, and 'G' repeats 2 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{11!}{3! \times 2! \times 2!} = \frac{39916800}{12} = 3326400 \]

### (vi) INTERMEDIATE

1. **Count the letters**: INTERMEDIATE has 12 letters.
2. **Identify repetitions**: The letter 'I' repeats 2 times, 'T' repeats 2 times, and 'E' repeats 3 times.
3. **Apply the formula**:
\[ \text{Permutations} = \frac{12!}{2! \times 2! \times 3!} = \frac{479001600}{12} = 39916800 \]

### Summary of Results:
- (i) APPLE: 60
- (ii) ARRANGE: 1260
- (iii) COMMERCE: 5040
- (iv) INSTITUTE: 45360
- (v) ENGINEERING: 3326400
- (vi) INTERMEDIATE: 39916800