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At the beginning of a party, each person present shook hands with all other people present and there were in all 45 handshakes. In the midst of the party, 2 persons left due to an emergency. Now, the number of males and females present in the party was equal. At the end, each female shook hands only with every female present and each male shook hands only with every male present. What is the total number of handshakes that took place at the party?

At the beginning of a party, each person present shook hands with all other people present and there were in all 45 handshakes. In the midst of the party, 2 persons left due to an emergency. Now, the number of males and females present in the party was equal. At the end, each female shook hands only with every female present and each male shook hands only with every male present. What is the total number of handshakes that took place at the party? Correct Answer 57

Calculation:

⇒ nC2 = 45

⇒ n = 10

⇒ Then, 2 people left the party and from the remaining people, number of males and females is equal.

⇒ So, there are 4 males and 4 females each.

⇒ At the end, people shake hands only with the persons of the same gender.

⇒ So, in the end 4C2 + 4C2 = 6 + 6 = 12 handshakes will take place.

⇒ Total number of handshakes = 45 + 12 = 57.

Additional Information

⇒ nCr = nCn - r

⇒ If nCr = nCk, then r = k or n - r = k

⇒ nCr + nCr-1 = n+1Cr

⇒ nCr = n/r  n-1Cr-1

⇒ nCr / nCr-1 = (n - r + 1)/r

⇒ If n is even nCr is greatest for r = n/2

⇒ If n is odd, nCr is greatest for r = (n - 1)/2, (n + 1)/2

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