What is the total number of handshakes at a couples-only party attended by several couples if there were 264 handshakes?

To determine the number of couples at the party and confirm the calculation leading to the total number of handshakes, it's important to understand how handshakes are counted in a scenario where only couples are present.

In a couples-only party, each couple consists of two people, and every person can shake hands with everyone else except for their partner. Therefore, if there are 'n' couples, there are '2n' individuals at the party. Since one person does not shake hands with their partner, each person can potentially shake hands with '2n - 1' other individuals.

However, since each handshake between two individuals is counted twice (once for each participant), the formula to calculate the total number of unique handshakes becomes:

Total handshakes = (2n * (2n - 2)) / 2.

This simplifies to:

Total handshakes = n(2n - 2) = 2n^2 - 2n.

Given that the total number of handshakes is 264, we can set up the equation:

2n^2 - 2n = 264.

Dividing through by 2 gives:

n^2 - n = 132,

which simplifies