There are p letters a, q letters b, r letters C.The number of ways of selecting k letters out of these if `p lt k lt q lt r` is

Options

A. `2^(n)`
B. `3^(n)`
C. `4^(n)`
D. `n 3^(n-1)`

Explanation

To solve the problem of selecting \( k \) letters from \( p \) letters \( a \), \( q \) letters \( b \), and \( r \) letters \( c \) under the condition \( p < k < q < r \), we will follow these steps:

### Step 1: Understand the constraints
We have:
- \( p \) letters \( a \)
- \( q \) letters \( b \)
- \( r \) letters \( c \)

The condition \( p < k < q < r \) implies that \( k \) must be greater than \( p \) and less than \( q \), and \( q \) must be less than \( r \). This means that \( k \) can take values from \( p + 1 \) to \( q - 1 \).

### Step 2: Determine the range for \( k \)
The possible values for \( k \) are:
- Minimum value: \( p + 1 \)
- Maximum value: \( q - 1 \)

The number of possible values for \( k \) is:
\[ (q - 1) - (p + 1) + 1 = q - p - 1 \]

### Step 3: Calculate the total selections
For each valid \( k \), we can select \( k \) letters from the total of \( p + q + r \) letters. The selection can be done in the following way:
1. Choose \( k \) letters from the total \( p + q + r \) letters.
2. The number of ways to choose \( k \) letters from \( p + q + r \) is given by the binomial coefficient \( C(p + q + r, k) \).

### Step 4: Summation over valid \( k \)
We need to sum the selections over all valid \( k \) values:
\[ \text{Total selections} = \sum_{k = p + 1}^{q - 1} C(p + q + r, k) \]

### Step 5: Final expression
The total number of ways to select \( k \) letters from \( p \) letters \( a \), \( q \) letters \( b \), and \( r \) letters \( c \) under the given constraints is:
\[ \sum_{k = p + 1}^{q - 1} C(p + q + r, k) \]