The combinatorial coefficients `""^(n – 1)C_(p)` denotes

Options

A. The number of ways in which n things of which p are alike and rest different can be arranged in a circle
B. The number of ways in which p different things can be selected out of n different thing if a particular thing is always excluded
C. Number of ways in which n alike balls can be distributed in p different boxes so that no box remains empty and each box can hold any number of balls
D. The number of ways in which (n – 2) white balls and p black balls can be arranged in a line if no two black balls are together, balls are all alike except for the colour

Answer

To solve the problem regarding the combinatorial coefficient \( C(n-1, p) \), we will analyze the meaning of this expression step by step.

### Step-by-Step Solution:

1. **Understanding the Notation**:
The notation \( C(n-1, p) \) or \( \binom{n-1}{p} \) represents the number of ways to choose \( p \) elements from a set of \( n-1 \) elements. This is known as a binomial coefficient.

**Hint**: Remember that \( C(n, r) \) is defined as \( \frac{n!}{r!(n-r)!} \).

2. **Identifying the Context**:
The problem asks us to interpret what \( C(n-1, p) \) denotes in a combinatorial context. This can relate to various scenarios such as arranging objects, selecting items, or distributing indistinguishable objects into distinguishable boxes.

**Hint**: Think about scenarios where you have a total of \( n-1 \) items and you need to select \( p \) from them.

3. **Analyzing Possible Conditions**:
- **Condition 1**: The number of ways to arrange \( n \) items where \( p \) are alike and the rest are different. This can often lead to a circular arrangement, which is more complex and may not directly relate to \( C(n-1, p) \).

**Hint**: Consider how circular arrangements differ from linear arrangements.

- **Condition 2**: The number of ways to choose \( p \) different items from \( n-1 \) items, which directly corresponds to \( C(n-1, p) \).

**Hint**: This condition directly matches the definition of the binomial coefficient.

- **Condition 3**: The distribution of \( n \) alike objects into \( p \) boxes where no box is empty. This scenario often uses the "stars and bars" theorem, which may not directly relate to \( C(n-1, p) \).

**Hint**: Recall the "stars and bars" theorem for distributing indistinguishable objects.

- **Condition 4**: Arranging \( n-2 \) alike balls and \( p \) black balls in a line with the restriction that no two black balls are adjacent. This is a combinatorial arrangement problem and may not directly yield \( C(n-1, p) \).

**Hint**: Think about how restrictions affect arrangements.

4. **Conclusion**:
After analyzing the conditions, we find that the second condition, which states the number of ways to choose \( p \) different items from \( n-1 \) items, directly corresponds to the combinatorial coefficient \( C(n-1, p) \).

**Final Answer**: The combinatorial coefficient \( C(n-1, p) \) denotes the number of ways to choose \( p \) items from \( n-1 \) items.