Question

At a party, everyone shook hands with everybody else. There were 66 handshakes. How many people were at the party? ​

Answer 1

Answer:

n(n-1)/2

there were 12 people

12(11)/2

132/2=66.

Step-by-step explanation:

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Answer 2

Answer:

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Step-by-step explanation:

There are 12 persons.

Let the number of persons= n

Number of handshakes by first person= n- 1

(As he cannot handshake with himself)

Number of handshakes by second person = n- 2

(As he has already handshaked with first person)

Number of handshakes by third person= n- 3

Number of handshakes by second last i.e. ( n- 1)th person= n- (n- 1)= 1

Number of handshakes by last i.e. nth person= n- n= 0

So,

Total number of handshakes=

    (n- 1)+ (n- 2)+ (n- 3)+ 1+ 0

=  0+ 1+ 2+ 3+ ...........(n- 1)

=  Sum of first (n- 1) terms

=  (n- 1)(n- 1+ 1)/2

=  n(n- 1)/ 2

So,

Total number of handshakes= n(n- 1)/2

According to question,

      n(n- 1)/ 2= 66

=>  n^2 - n= 132

=>  n^2- n- 132= 0

=>  n^2- 12n+ 11n- 132= 0

=>  n(n- 12) + 11(n- 12)= 0

=>  (n- 12)(n+ 11) = 0

=>  n= 12   or n= - 11(which is not possible as no. Of persons cannot be negative)

=> n= 12

Hence, there are 12 persons in the party.

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