How many combinations of 4 letters can be made of the letters of the word ‘JAIPUR’ ?

Answer

To find the number of combinations of 4 letters that can be formed from the letters of the word "JAIPUR", we can follow these steps:

### Step 1: Identify the total number of letters
The word "JAIPUR" consists of 6 distinct letters: J, A, I, P, U, R.

### Step 2: Determine the number of letters to choose
We need to form combinations of 4 letters from these 6 letters.

### Step 3: Use the combination formula
The formula for combinations is given by:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

In our case, \( n = 6 \) (the total letters) and \( r = 4 \) (the letters we want to choose).

### Step 4: Substitute values into the formula
Substituting the values into the combination formula:
\[ C(6, 4) = \frac{6!}{4!(6 - 4)!} = \frac{6!}{4! \cdot 2!} \]

### Step 5: Calculate the factorials
Now, we calculate the factorials:
- \( 6! = 6 \times 5 \times 4! \)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 2! = 2 \times 1 = 2 \)

### Step 6: Substitute the factorial values back into the equation
Now we can simplify:
\[ C(6, 4) = \frac{6 \times 5 \times 4!}{4! \cdot 2} = \frac{6 \times 5}{2} \]

### Step 7: Simplify the expression
Now, we simplify:
\[ C(6, 4) = \frac{30}{2} = 15 \]

### Conclusion
Thus, the number of combinations of 4 letters that can be made from the letters of the word "JAIPUR" is **15**.

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