Question:
Everyone in the room handshakes with everyone else. If the total number of handshakes is 66, then the number of persons in the room is...
Everyone in the room handshakes with everyone else. If the total number of handshakes is 66, then the number of persons in the room is...
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Each handshake is a choice of 2 persons from n, so n(n-1)/2 equals 66.
Updated On: Nov 12, 2026
- \(10\)
- \(6\)
- \(11\)
- \(12\)
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Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
Every handshake involves a pair, so the total is \(^{n}C_2=\dfrac{n(n-1)}{2}\).
Step 2: Calculation
\(\dfrac{n(n-1)}{2}=66\Rightarrow n^2-n-132=0\)
\((n-12)(n+11)=0\)
Since \(n>0\), \(n=12\).
Step 3: Compare the options
The value 12 is the last option.
Final Answer:
There are 12 persons in the room. \[ \boxed{\text{(D) }12} \]
Every handshake involves a pair, so the total is \(^{n}C_2=\dfrac{n(n-1)}{2}\).
Step 2: Calculation
\(\dfrac{n(n-1)}{2}=66\Rightarrow n^2-n-132=0\)
\((n-12)(n+11)=0\)
Since \(n>0\), \(n=12\).
Step 3: Compare the options
The value 12 is the last option.
Final Answer:
There are 12 persons in the room. \[ \boxed{\text{(D) }12} \]
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