4. If 15 people greet each other at a meeting by shaking hands with one another, how many handshakes will take place? [ 5 pts$$ ] $$

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To find the total number of handshakes that occur when 15 people greet each other, we can use the formula for combinations. Each handshake involves a unique pair of people, and we want to find the number of ways to choose 2 people from a group of 15.

The formula for combinations is given by:

$$
C(n, r) = \frac{n!}{r!(n-r)!}
$$

where $$ n $$ is the total number of people, $$ r $$ is the number of people in each handshake (which is 2), and $$ ! $$ denotes factorial.

In this case, $$ n = 15 $$ and $$ r = 2 $$:

$$
C(15, 2) = \frac{15!}{2!(15-2)!} = \frac{15!}{2! \cdot 13!}
$$

This simplifies to:

$$
C(15, 2) = \frac{15 \times 14}{2 \times 1} = \frac{210}{2} = 105
$$

Thus, the total number of handshakes that will take place is **105**.
The total number of handshakes is 105.

4. If 15 people greet each other at a meeting by shaking hands with one another, how many handshakes will take place? [ 5 pts\( ] \)

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