Aman Verma and Mini Mathur jointly host a TV programme in which one day n guests attended the show. There each guest shakes hands with every other guest and each guest has to shake hand with each of the hosts. If there happens to be total 65 shake hands, find the number of guests attended the show.

Options

A. 13
B. 14
C. 10
D. 9

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Handshake Scenario
Each guest shakes hands with every other guest and also with both hosts. If there are \( n \) guests, we need to calculate the total number of handshakes.

### Step 2: Handshakes Among Guests
The number of handshakes among \( n \) guests can be calculated using the formula for combinations:
\[ \text{Handshakes among guests} = \frac{n(n-1)}{2} \]
This formula arises because each guest shakes hands with \( n-1 \) other guests, and we divide by 2 to avoid double counting.

### Step 3: Handshakes with Hosts
Each guest shakes hands with both hosts. Therefore, the number of handshakes between guests and hosts is:
\[ \text{Handshakes with hosts} = 2n \]
This is because each guest shakes hands with both Aman and Mini.

### Step 4: Total Handshakes Equation
The total number of handshakes is the sum of handshakes among guests and handshakes with hosts:
\[ \frac{n(n-1)}{2} + 2n = 65 \]

### Step 5: Clear the Fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ n(n-1) + 4n = 130 \]

### Step 6: Rearrange the Equation
Rearranging gives us:
\[ n^2 - n + 4n - 130 = 0 \]
This simplifies to:
\[ n^2 + 3n - 130 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ (n + 13)(n - 10) = 0 \]

### Step 8: Solve for \( n \)
Setting each factor to zero gives us:
1. \( n + 13 = 0 \) → \( n = -13 \) (not possible since \( n \) must be positive)
2. \( n - 10 = 0 \) → \( n = 10 \)

### Conclusion
Thus, the number of guests who attended the show is:
\[ \boxed{10} \]

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