Question

At a party, everyone shook hands with everybody else. There were 253 handshakes. How many people were at the party?

• A. 23
• B. 22
• C. 21
• D. 20

Answer 1

The correct option is A 23

Well, let’s try posing a hypothetical scenario to understand the situation, If there were 4 people, how many handshakes would there be?
P, Q, R, and S: each letter denotes a unique person Person
P could shake hands with Q, R and S.
Person Q could shake hands with R, and S.
Do note that, since we have already kept track of P shaking hands with Q, let’s not recount it here. Person R could shake hands with only S.

In total, there are 3+2+1 = 6 unique handshakes.
If you look closely, the number of handshakes is equal to the sum of natural numbers up to n-1, where n is the total number of people. Huh, that’s interesting…

Now, the handshake problem can simply be solved. The sum of natural numbers up to n is given by

Sum=n(n+1)/2
For sum upto n-1, the above equation will reduce to,
S=(n-1)(n-1+1)/2
S=n(n-1)/2

Where, S represents the total number of unique handshakes.
We can now return to the actual question, 253 unique handshakes have happened and we have to find the number of individuals at the part.
Here, S= 253.
$\Rightarrow$ n(n-1)/2=253
$\Rightarrow$ n(n-1)=506
$\Rightarrow n^2-n-506$,
Solving the quadratic equation, we get, n=23 or n=-22.
Since number of individuals cannot be negative, the correct answer is 23.