Question

After a get-together every person present shakes the hand of every other person. If there were 105 handshakes in all, how many persons were present in the party?

• A. 15
• B. 14
• C. 13
• D. 16

Answer 1

The Correct Option is A

\(\text{Total handshakes} = \frac{n(n-1)}{2}\)

where \(n\) is the number of people.
\(\frac{n(n-1)}{2} = 105\)
 Solve for \(n\):
 Multiply both sides by 2 to eliminate the fraction:
 \(n(n-1) = 210\)
 Now, solve the quadratic equation:
 \(n^2 - n - 210 = 0\)
Solve using the quadratic formula:
 \(n = \frac{-(-1) \pm \sqrt{(-1)^2 + 4 \cdot 1 \cdot 210}}{2 \cdot 1} = \frac{1 \pm \sqrt{841}}{2} = \frac{1 \pm 29}{2}\)
 \(n = \frac{1 + 29}{2} = 15\)
 \(n = \frac{1 - 29}{2} = -14\) (not valid)

There were 15 persons present at the party.