Question

In a conference of \[8\] persons, if each person shakes hands with the other one only, then how many total number of shake hands will there be?
A \[64\]
B \[56\]
C \[49\]
D \[28\]

Answer 1

Hint: To find the total number of shake hands, use the combination concept. A mathematical technique to find the number of possible arrangements in a set of items. In the combination order doesn’t matter.

Formula used: \[{}^n{C_r} = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}\]
Here, n is the number of items in the set, r is the number of ways of arrangement.

Complete step by step solution: Given that, there are \[8\] persons in the conference. Each person shakes hands with the other one. No one can shake hands with self. To find the total number of shake hands use a combination formula.
Here, The value of \[n\] is \[8\] as we have \[8\] persons and the value of \[r\] is \[2\] as a handshake is between two hands. Now apply formula as follows.
\[ \Rightarrow \dfrac{{8!}}{{\left( {8 - 2} \right)!2!}}\]
Simplify further as follows.
\[\begin{array}{l} \Rightarrow \dfrac{{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8}}{{1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 1 \times 2}}\\ \Rightarrow \dfrac{{7 \times 8}}{2}\\ \Rightarrow 7 \times 4\\ \Rightarrow 28\end{array}\]
Hence, the total number of shake hands will be \[28\].

Thus, Option (D) is correct.

Note: Students often make the mistake of considering \[8\] persons hand shakes with other persons so, total hand shakes are \[8 \times 7 = 56\] which is wrong.