Every person in a room , shake hands with every other person . The total number of handshakes it 45. The number of person in the room is .

Options

A. 5
B. 10
C. 15
D. 20

Answer

To solve the problem of finding the number of people in a room where every person shakes hands with every other person resulting in a total of 45 handshakes, we can follow these steps:

### Step 1: Understand the Handshake Formula
When each person shakes hands with every other person, the total number of handshakes can be represented by the combination formula \( C(n, 2) \), where \( n \) is the number of people. The formula for combinations is given by:
\[ C(n, 2) = \frac{n(n-1)}{2} \]

### Step 2: Set Up the Equation
Given that the total number of handshakes is 45, we can set up the equation:
\[ \frac{n(n-1)}{2} = 45 \]

### Step 3: Eliminate the Fraction
To eliminate the fraction, multiply both sides of the equation by 2:
\[ n(n-1) = 90 \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us a standard quadratic equation:
\[ n^2 - n - 90 = 0 \]

### Step 5: Apply the Quadratic Formula
To solve for \( n \), we can use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -90 \). Plugging in these values:
\[ n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-90)}}{2 \cdot 1} \]
\[ n = \frac{1 \pm \sqrt{1 + 360}}{2} \]
\[ n = \frac{1 \pm \sqrt{361}}{2} \]
\[ n = \frac{1 \pm 19}{2} \]

### Step 6: Calculate Possible Values for \( n \)
Calculating the two possible values:
1. \( n = \frac{1 + 19}{2} = \frac{20}{2} = 10 \)
2. \( n = \frac{1 - 19}{2} = \frac{-18}{2} = -9 \)

### Step 7: Determine the Valid Solution
Since the number of people cannot be negative, we discard \( n = -9 \). Thus, the only valid solution is:
\[ n = 10 \]

### Conclusion
The number of people in the room is \( \boxed{10} \).

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