Find the number of words which can be formed using all the letters of the word 'INSTITUTION' which start with consonant.

Explanation

To find the number of words that can be formed using all the letters of the word "INSTITUTION" that start with a consonant, we will follow these steps:

### Step 1: Identify the letters and their counts
The word "INSTITUTION" consists of the following letters:
- I: 3 times
- N: 2 times
- S: 1 time
- T: 3 times
- U: 1 time
- O: 1 time

Total letters = 11

### Step 2: Identify the consonants
The consonants in "INSTITUTION" are:
- N, S, T

Counting the consonants:
- N: 2 times
- S: 1 time
- T: 3 times

Total consonants = 6 (N, N, S, T, T, T)

### Step 3: Choose the first letter (which must be a consonant)
For the first letter, we can choose from the 6 consonants.

### Step 4: Calculate the arrangements of the remaining letters
After choosing a consonant for the first position, we will have 10 letters left to arrange. The arrangement of these letters will depend on their counts.

### Step 5: Calculate the total arrangements
The total arrangements of the remaining 10 letters will be calculated using the formula for permutations of multiset:

\[ \text{Total arrangements} = \frac{10!}{\text{(count of I)!} \times \text{(count of N)!} \times \text{(count of T)!}} \]

Where:
- \(10!\) is the factorial of the total letters left.
- The counts of I, N, and T are taken into account since they are repeating.

### Step 6: Substitute the values
Substituting the values:
- Count of I = 3
- Count of N = 2
- Count of T = 3

So, the formula becomes:

\[ \text{Total arrangements} = \frac{10!}{3! \times 2! \times 3!} \]

### Step 7: Calculate the factorials
Now, we calculate the factorials:
- \(10! = 3628800\)
- \(3! = 6\)
- \(2! = 2\)

### Step 8: Calculate the total arrangements
Now substituting the values into the formula:

\[ \text{Total arrangements} = \frac{3628800}{6 \times 2 \times 6} = \frac{3628800}{72} = 50400 \]

### Step 9: Multiply by the number of choices for the first letter
Since we have 6 choices for the first letter (the consonants), we multiply the total arrangements by 6:

\[ \text{Total words starting with a consonant} = 6 \times 50400 = 302400 \]

### Final Answer
The total number of words that can be formed using all the letters of the word "INSTITUTION" that start with a consonant is **302400**.

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