Problem Analysis: Total Handshakes

This problem involves calculating the total number of unique pairs that can be formed from a group of 40 individuals, where each pair represents a handshake. Since the order of people in a handshake does not matter (Person A shaking Person B's hand is the same handshake as Person B shaking Person A's hand), this is a combination problem.

Applying the Combination Formula

The number of combinations of choosing k items from a set of n items is given by the formula:

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

For the handshake problem, we are choosing groups of 2 people (k=2) from a total of 40 people (n=40). The formula simplifies to:

$C(n, 2) = \frac{n(n-1)}{2}$

Calculation for 40 Persons

Given n = 40 persons:

• Substitute the value of n into the simplified formula:

  $C(40, 2) = \frac{40 \times (40 - 1)}{2}$

• Perform the subtraction:

  $C(40, 2) = \frac{40 \times 39}{2}$

• Simplify the expression:

  $C(40, 2) = \frac{1560}{2}$

• Calculate the final result:

  $C(40, 2) = 780$

Therefore, the total number of handshakes will be 780.