Number of four letter words can be formed using the letters of word VIBRANT if letter V is must included, are :

Options

A. 840
B. 480
C. 120
D. 240

Answer

To solve the problem of how many four-letter words can be formed using the letters of the word "VIBRANT" with the condition that the letter 'V' must be included, we can follow these steps:

### Step 1: Identify the letters available
The letters in the word "VIBRANT" are V, I, B, R, A, N, T. This gives us a total of 7 letters.

### Step 2: Fix the position of 'V'
Since 'V' must be included in every four-letter word, we can fix one position for 'V'. This means we will have one letter 'V' and we need to choose 3 more letters from the remaining letters.

### Step 3: Count the remaining letters
After fixing 'V', the remaining letters available to choose from are I, B, R, A, N, T. This gives us a total of 6 letters to choose from.

### Step 4: Choose 3 letters from the remaining 6
We need to select 3 letters from the 6 remaining letters. The number of ways to choose 3 letters from 6 can be calculated using the combination formula:

\[ \text{Number of ways to choose 3 letters} = \binom{6}{3} \]

Calculating this:

\[ \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \]

### Step 5: Arrange the chosen letters
Now, we have chosen 3 letters along with the fixed letter 'V'. This gives us a total of 4 letters (V + 3 chosen letters). The number of ways to arrange these 4 letters is given by:

\[ \text{Number of arrangements} = 4! = 24 \]

### Step 6: Calculate the total number of four-letter words
To find the total number of four-letter words, we multiply the number of ways to choose the letters by the number of arrangements:

\[ \text{Total number of words} = \binom{6}{3} \times 4! = 20 \times 24 = 480 \]

### Final Answer
Thus, the total number of four-letter words that can be formed using the letters of the word "VIBRANT" with 'V' included is **480**.

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