# Handshake problem

#### evinda

##### Well-known member
MHB Site Helper
Hello !!!
Could you help me at the exercise below?
Suppose that n couples are at a party.
If every person at the party shake hands with any other person except from his partner, how many handshakes will have been exchanged?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Hello !!!
Could you help me at the exercise below?
Suppose that n couples are at a party.
If every person at the party shake hands with any other person except from his partner, how many handshakes will have been exchanged?
Hi evinda!

Suppose we have 3 couples, say persons A, a, B, b, C, and c.
How many hands does A shake?
How many handshakes are there in total?
Can you generalize?

#### evinda

##### Well-known member
MHB Site Helper
"A" and "a" shake 4 hands,"B" and "b" shake 2 hands.Can you give me a hint how to find the general formula,because I have stuck?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
"A" and "a" shake 4 hands,"B" and "b" shake 2 hands.Can you give me a hint how to find the general formula,because I have stuck?
Actually, "A" and "a" shake 4 hands, "B" and "b" shake 4 hands, and "C" and "c" shake 4 hands.
So there are 6 x 4 times that someone shakes a hand.
Since it takes 2 persons to do a handshake, we should divide the total number by 2.
That means that the number of handshakes is 6 x 4 / 2 = 12.

Generalize?

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Here is an illustration.

#### evinda

##### Well-known member
MHB Site Helper
Is it $$\frac{n\cdot (n-2)}{2}$$ ,where n the number of persons that are at the party ?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Is it $$\frac{n\cdot (n-2)}{2}$$ ,where n the number of persons that are at the party ?
Yep!

Btw, in your problem statement, n was supposed to be the number of couples.
I'd advise against mixing up the meaning of symbols.
Your number of handshakes is $$\frac{2n\cdot (2n-2)}{2}$$, where $n$ is the number of couples.

#### evinda

##### Well-known member
MHB Site Helper
Nice!!!Thank you very much!!!!!