Every body in a room shakes hands with everybody else. The total number of handshakes is 21. The total number of persons in the room is

Options

A. (a) 6
B. (b) 7
C. (c) 8
D. (d) 9

Answer

To solve the problem of finding the total number of persons in a room where every person shakes hands with every other person, and the total number of handshakes is 21, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Each handshake involves 2 people. If there are \( N \) people in the room, the number of unique handshakes can be calculated using combinations.

2. **Using the Combination Formula**:
- The number of ways to choose 2 people from \( N \) is given by the combination formula:
\[ \binom{N}{2} = \frac{N!}{(N-2)! \cdot 2!} \]
- This simplifies to:
\[ \binom{N}{2} = \frac{N(N-1)}{2} \]

3. **Setting Up the Equation**:
- According to the problem, the total number of handshakes is 21. Therefore, we can set up the equation:
\[ \frac{N(N-1)}{2} = 21 \]

4. **Eliminating the Fraction**:
- To eliminate the fraction, multiply both sides by 2:
\[ N(N-1) = 42 \]

5. **Rearranging the Equation**:
- Rearranging gives us a quadratic equation:
\[ N^2 - N - 42 = 0 \]

6. **Factoring the Quadratic**:
- We need to factor the quadratic equation. We are looking for two numbers that multiply to -42 and add to -1. The numbers -7 and 6 work:
\[ (N - 7)(N + 6) = 0 \]

7. **Finding the Solutions**:
- Setting each factor to zero gives us:
\[ N - 7 = 0 \quad \Rightarrow \quad N = 7 \]
\[ N + 6 = 0 \quad \Rightarrow \quad N = -6 \]

8. **Interpreting the Results**:
- Since \( N \) represents the number of persons, it cannot be negative. Therefore, we discard \( N = -6 \).

9. **Conclusion**:
- The total number of persons in the room is:
\[ N = 7 \]

### Final Answer:
The total number of persons in the room is **7**.

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