handshake math question?

Question

there were a total of 78 handshakes. if each person shook hands once with each of the other people, how many people were at the party?

Answer 1

There were 13 people.

A handshake can be mathematically compared to pairing couples. So, suppose there were 'n' number of people at the party & they all shook hands with each other, means like they were paired together, in this case total no. of handshakes can be calculated using Combinations formula:

n C 2 = 78

n*(n-1) / 2! = 78

n*(n-1) = 78*2 = 156

n^2 - n -156 = 0

Solving this quadrating eqn:

n = { 1 +/- sqrt(1 - (4*(-156))) } / 2

n = { 1 +/- sqrt(625) } / 2

n = { 1 +/- 25 } / 2

n = 26/2 (or) n = -24/2

n=13 (or) n=(-12)

Since 'n' is no. of people & cannot be negative, we have to go with n=13

Hence, there were 13 people at the party

Answer 2

The way to solve this depends on what grade you are in. If you are grade 7 or 8, the methods given so far are too complicated.

In a "shaking hands" question, usually it is stated that there are a certain number of people, each shakes hands once, how many handshakes were there. (the question can also be teams in a tournament or something like that too). You solve that by adding up the sequence of numbers, starting at one less than the total number of people. eg. 4 people --- 3+2+1= 6 handshakes

so to solve this question, you would know that you had to keep adding up numbers till you got to 78

You could also solve this by "guess and check" method

eg guess 10 people

9+8+7+6+5+4+3+2+1=45 --- too few

guess 15

14+13+12+11+10+9+8+7+6+5+4+3+2+1=95 too many

so guess inbetween , say 13

12+11+10+9+8+7+6+5+4+3+2+1=78 correct

Answer 3

THE ANSWER IS 13. Anyone who wrote anything else is an idiot and should not post incorrect answers.

The correct way to get this answer is to realize that if there are n people, person 1 will shake hands with n-1 people, as they cannot shake hands with themself. Person 2 will shake hands with n-2 people, since they cannot shake hands with themselves and they already shaked hands with person 1.

The amount of people that there are, n, can be expressed by a summation.

The amount of handshakes=the summation of k from k=1 to n-1.

So 78=1+2+3+4+5+6+7+8+9+10+11+12

if n-1=12 then n=13.

This is the correct way to approach this problem as sometimes the numbers can get very large and adding all of them would be impractical.

To solve the summation of k from k=1 to r the formula:
r(r+1)/2

For the summation about 78=summation of k from k=1 to n-1 the summation becomes (n-1)(n)/2.

So 78=n(n-1)/2 so
n^2-n=156

Solving this you get n=13.

This summation summing k from k=1 to r is an arithmetic series and you must memorize this. It is a very common series.

Answer 4

let total number of ppl = N

for each one person, number of handshakes = N-1 times

for N ppl, number of handshakes = N* (N-1)

as above total number of handshakes include A handshake with B and B handshake with A, count twice..

so actual total number of handshakes = N* (N-1) / 2 =78

solve above equation

N^2-N=78*2

N^2-N-156=0
=> (N-13)*(N+12)=0 so N=13

Answer 5

13

Answer 6

79, this is more of a thinking issue

2 people shake hands = 1 handshake
3 people at the party, a, b, c. a shakes with b and c, so 2 handshakes. You see you will always have 1 less handshake than people.

Answer 7

The Sum on (n-i) for i=1 to (n) equals 78.
This can be resolved like n(n-1)/2 =78
so n2-n-156 =0 ...
(n-13)(n+12)=0 ...
n=13 or n=-12

so the number of people is 13.

Answer 8

It took 2 people to make each handshake, so 156 people were involved in them . Each person shakes hands with one less that the total number of people (doesn't shake her own hand).

So if there were n people, n(n-1) = 156

Solve for n either by multiplying out, or by simply thinking of two numbers that are one apart whose product is 156.

Answer 9

the formula n(n-1)/2 results in the sum.
n(n-1)/2=78
n^2 - n = 156
n^2 -n - 156 = 0
(n - 13) (n + 12) = 0
n = 13, -12, but there can't be a negative number of people so 13 people.

got the formula wrong. correction. my bad

Answer 10

there were x people
first person shook hands with (x-1) people [all but themselves]
next person shook hands with (x-2) people [all but themselves and the first person]
etc

so you're looking for

1+2+3+...+(x-1)+x=78

let x be an even number (if i'm wrong, we'll correct), then

...+((x-1)+2)+(x+1)=78

(x/2) * (x+1) = 78

x^2 + x = 156
x^2 + x -156 = 0
(x+13)(x-12)=0
x=12 or -13 (can't have -13 people), so there were 12 people.