Quote:
How many guests were present at the party?
(1) There were 45 handshakes if the guests shook hands with each other and every 2 guests shook hands exactly once
(2) If there were half as many guests present, they could sit in a row in 120 different ways
Question: Number of guests in Party = ?STatement (1) There were 45 handshakes if the guests shook hands with each other and every 2 guests shook hands exactly onceTotal Number of handshakes with n people can be written in three ways
Method 1: if total people = n then First person will have n-1 people to shake hands with i.e. his handshakes = n-1
Second person has shook hand with first so second will shake hands with n-2 people and similarly the second last person will have 1 person to shake hands with
i.e. Total handshakes = 1+2+3+4+....+(n-1) = 45
i.e. n = 10
Method 2: Everyone of n people has (n-1) other people to shake hands with so total handshakes = n*(n-1)
but every handshake has been counted twice e.g. first shakes hand with third when his handshakes are counted and third person shakes hand with first when his handshakes are counted
hence total unique handshakes = n*(n-1)/2 = 45 i.e. n = 10
Method 3: We need two persons to cause one handshake
i.e. Total handshakes = total ways of choosing 2 persons out of n \(= nC_2 = 45\)
i.e. n = 10
SUFFICIENT
Statement (2) If there were half as many guests present, they could sit in a row in 120 different waysIf total guests = n
then half the guests = n/2
total ways of arranging n/2 people in a row = (n/2)! = 120 (given)
but 5! = 120
i.e. n/2 = 5 i.e. n = 10
SUFFICIENT
Answer: Option D