# Solve the Handshake Problem: n Couples at a Party

• MHB
• evinda
In summary, we have discussed the problem of determining the number of handshakes that occur at a party with n couples, where every person shakes hands with someone else except their partner. The general formula for this is \frac{2n\cdot (2n-2)}{2}, where n is the number of couples.
evinda
Gold Member
MHB
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?

evinda said:
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?

"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?

evinda said:
"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?

Here is an illustration.

Is it $$\frac{n\cdot (n-2)}{2}$$ ,where n the number of persons that are at the party ?

evinda said:
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.

Nice!Thank you very much! ;)

## 1. How many handshakes will occur at a party with n couples?

The number of handshakes can be calculated using the formula n(n-1)/2, where n is the number of couples. This is because each person will shake hands with every other person once, and there are n couples, so there are n(n-1) total possible handshakes. However, this formula only applies if everyone shakes hands with everyone else, which may not be the case at a party.

## 2. Is it possible for two people to shake hands more than once at a party with n couples?

Yes, it is possible for two people to shake hands more than once at a party with n couples. This can happen if they accidentally shake hands multiple times or if they intentionally shake hands multiple times throughout the party. However, the formula for calculating the number of handshakes assumes that each person will only shake hands with each other person once.

## 3. Can the handshake problem be solved for an odd number of couples?

Yes, the handshake problem can be solved for an odd number of couples. The formula for calculating the number of handshakes still applies, as it is based on the number of pairs of people, not the total number of people. However, there may be some people who do not have a partner to shake hands with, depending on how the couples are arranged.

## 4. How does the number of handshakes change as the number of couples increases?

The number of handshakes increases exponentially as the number of couples increases. For example, with 1 couple, there is only 1 handshake. With 2 couples, there are 2 handshakes. But with 10 couples, there are 45 handshakes, and with 100 couples, there are 4,950 handshakes. This is due to the formula n(n-1)/2, where n represents the number of couples.

## 5. Can the handshakes at a party with n couples be visualized?

Yes, the handshakes at a party with n couples can be visualized using a graph or network diagram. Each person can be represented as a node, and each handshake can be represented as an edge connecting two nodes. This can help to better understand the handshakes and see patterns or clusters within the group. There are also online tools available to automatically generate such visualizations for different numbers of couples.

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