Answer

Correct Answer : d ) 90

Finding the number of Handshakes between persons at a party problem is one of the most famous mathematical problems. Let us see how to approach such a problem in the clearest and easy way.

Let us assume a party with n people is running.

Every person decides to shake hands with every other person.

Now, any person will be shaking hands with (n - 1) persons only as a person cannot shake hands with himself.

So, all n persons will shake hands (n - 1) times.

Now, think about a case where Mr. John and Mr. Narendra are two persons among 'n' persons in the party. While we counted (n - 1) handshakes for Mr. John, it included the handshake of Mr. John and Mr. Narendra also. Now, counting (n - 1) handshakes for Mr. Narendra, it will also include the handshake of Mr. John and Mr. Narendra.

So, in our counting of (n - 1) handshakes for n persons, one handshake is counted twice.

So, the Total number of handshakes at the party will be n * (n - 1)/ 2

Now, coming back to the question,

Total number of persons = 10.

Total number of handshakes = 10 * 9/ 2 = 45.

It is given that handshakes took place at the start as well as end of the party, so total handshakes at the party = 2 * 45 = 90.

Hence, (d) is the correct answer.