In a handshake problem with a limit on outcomes, what combination represents "at most" that limit?

In combinatorial problems, when considering handshakes or pairwise interactions among ( n ) individuals, the goal is often to determine how many unique interactions can occur. The phrase "at most" implies that we are interested in the maximum number of distinct pairs from a set of individuals.

The representation ( nC2 ) refers to the number of ways to choose 2 individuals from a total of ( n ) individuals. This is calculated using the combination formula which is defined as:

[

nCk = \frac{n!}{k!(n-k)!}

]

For ( nC2 ), this specifically translates to:

[

nC2 = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2}

]

This formula provides the total number of unique pairings possible when each individual can handshake with every other individual exactly once. Thus, ( nC2 ) directly corresponds to the scenario of counting handshakes, making it the correct choice for representing "at most" that limit in the context of handshakes.

Other combinations like ( nC1 ), ( nC3 ), and ( nC4 )