How many different words beginning and ending with a consonant can be made out of the letters of the word ' EQUATION' ?

Answer

To solve the problem of how many different words beginning and ending with a consonant can be made out of the letters of the word 'EQUATION', we can follow these steps:

### Step 1: Identify the letters in the word 'EQUATION'
The word 'EQUATION' consists of 8 letters:
- Vowels: E, U, A, I, O (5 vowels)
- Consonants: Q, T, N (3 consonants)

### Step 2: Determine the positions for consonants
Since we need the words to begin and end with a consonant, we can denote the positions as follows:
- First position: Consonant
- Last position: Consonant

### Step 3: Choose consonants for the first and last positions
We have 3 consonants (Q, T, N). We need to select 2 consonants for the first and last positions. The number of ways to choose 2 consonants from 3 and arrange them in the first and last positions is given by:
\[ 3P2 = 3!/(3-2)! = 3 \times 2 = 6 \]

### Step 4: Determine the remaining letters
After placing the consonants in the first and last positions, we have used 2 consonants, leaving us with:
- 1 remaining consonant
- 5 vowels

This gives us a total of 6 letters (1 consonant + 5 vowels) that can fill the middle positions.

### Step 5: Arrange the remaining letters
The number of ways to arrange these 6 letters (1 consonant + 5 vowels) is given by:
\[ 6! = 720 \]

### Step 6: Calculate the total number of arrangements
Now, we multiply the number of ways to arrange the consonants in the first and last positions by the number of arrangements of the remaining letters:
\[ \text{Total arrangements} = 6 \times 720 = 4320 \]

### Final Answer
Thus, the total number of different words that can be formed from the letters of 'EQUATION' that begin and end with a consonant is **4320**.

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