How many different 4-letter words can be formed with the letters of the word ‘JAIPUR’ when A and I are always to be included ?

Answer

To solve the problem of how many different 4-letter words can be formed with the letters of the word ‘JAIPUR’ when A and I are always included, we can follow these steps:

### Step 1: Identify the letters in 'JAIPUR'
The letters in the word 'JAIPUR' are J, A, I, P, U, R. This gives us a total of 6 letters.

### Step 2: Include A and I
Since A and I must always be included in the 4-letter words, we will select A and I first. This leaves us with the letters J, P, U, and R to choose from for the remaining 2 letters.

### Step 3: Choose 2 additional letters
We need to choose 2 more letters from the remaining 4 letters (J, P, U, R). The number of ways to choose 2 letters from 4 can be calculated using the combination formula:

\[ \text{Number of ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

Here, \( n = 4 \) (the remaining letters) and \( r = 2 \) (the letters we want to choose).

\[ \text{Number of ways} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 4: Arrange the selected letters
Now, we have selected A, I, and 2 additional letters (let's denote them as X and Y). The total number of letters we now have is 4 (A, I, X, Y). The number of ways to arrange these 4 letters is given by \( 4! \):

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 5: Calculate the total arrangements
To find the total number of different 4-letter words, we multiply the number of ways to choose the additional letters by the number of arrangements of the 4 letters:

\[ \text{Total arrangements} = \text{Ways to choose letters} \times \text{Arrangements of letters} = 6 \times 24 = 144 \]

### Final Answer
Thus, the total number of different 4-letter words that can be formed from the letters of the word ‘JAIPUR’, including A and I, is **144**.

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