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

Guest Feb 25, 2015

#3**+5 **

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

Mmm

Let there be k people at the party.

The first person shook with k-1 people.

the second with a further k-2 people

the kth person did not shake with anyone new.

So the number of handshakes was

1+2+3+.....+(k-1)

this is the sum of an AP S=n/2(a+L) = $$\frac{k-1}{2}(1+(k-1))=\frac{k(k-1)}{2}$$

so

$$\\\frac{k(k-1)}{2}=66\\\\

k(k-1)=132\\\\

k^2-k-132=0\\\\$$

$${{\mathtt{k}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{k}}{\mathtt{\,-\,}}{\mathtt{132}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{k}} = {\mathtt{12}}\\

{\mathtt{k}} = -{\mathtt{11}}\\

\end{array} \right\}$$

Obviously there is not a neg number of people so there must be 12 people.

Melody
Feb 25, 2015

#2**+5 **

We can solve this by this "formula"

n(n-1)/ 2 = 66 multiply by 2 on each side

n(n-1) = 132 simplify and rearrange

n^2 - n - 132 = 0 factor

(n - 12) (n+11) = 0 and setting each factor to 0, we have that n = 12 and n = -11

Reject the negative

n = 12 people

We can see this easily

The 12th person shakes 11 hands

The 11th person shakes 10 hands, etc.

So...... the sum of

11 + 10 + 9 +......+ 3 + 2 + 1 = 66

CPhill
Feb 25, 2015

#3**+5 **

Best Answer

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

Mmm

Let there be k people at the party.

The first person shook with k-1 people.

the second with a further k-2 people

the kth person did not shake with anyone new.

So the number of handshakes was

1+2+3+.....+(k-1)

this is the sum of an AP S=n/2(a+L) = $$\frac{k-1}{2}(1+(k-1))=\frac{k(k-1)}{2}$$

so

$$\\\frac{k(k-1)}{2}=66\\\\

k(k-1)=132\\\\

k^2-k-132=0\\\\$$

$${{\mathtt{k}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{k}}{\mathtt{\,-\,}}{\mathtt{132}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{k}} = {\mathtt{12}}\\

{\mathtt{k}} = -{\mathtt{11}}\\

\end{array} \right\}$$

Obviously there is not a neg number of people so there must be 12 people.

Melody
Feb 25, 2015