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!!!!!