Understanding the Problem: Handshakes and Hugs

This problem requires us to calculate the total number of interactions, specifically handshakes and hugs, occurring at a tea party. The interactions depend on the gender of the people involved. We are given the number of males and females and the conditions for each type of interaction.

• Number of males present: 6
• Number of females present: 4
• A handshake occurs between two people of the same gender.
• A hug occurs between two people of opposite genders.

Calculating Handshakes

Handshakes happen only between individuals of the same gender. We need to find the number of pairs that can be formed within the group of males and within the group of females. Since the order of people in a handshake does not matter, we use combinations. The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items to choose from, and $k$ is the number of items to choose.

Handshakes among Males

There are 6 males, and a handshake requires 2 people. The number of possible handshakes among males is calculated using combinations: $ C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4!}{ (2 \times 1) \times 4!} = \frac{30}{2} = 15 $ There are 15 handshakes among the males.

Handshakes among Females

Similarly, there are 4 females, and a handshake requires 2 people. The number of possible handshakes among females is: $ C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3 \times 2!}{ (2 \times 1) \times 2!} = \frac{12}{2} = 6 $ There are 6 handshakes among the females.

Total Handshakes

The total number of handshakes is the sum of handshakes among males and handshakes among females: $ \text{Total Handshakes} = (\text{Handshakes among males}) + (\text{Handshakes among females}) $ $ \text{Total Handshakes} = 15 + 6 = 21 $

Calculating Hugs

Hugs occur between individuals of opposite genders. This means each hug involves exactly one male and one female. To find the total number of hugs, we can use the multiplication principle: multiply the number of choices for males by the number of choices for females.

Number of ways to choose 1 male from 6 = 6. Number of ways to choose 1 female from 4 = 4.

The total number of hugs is calculated as follows: $ \text{Total Hugs} = (\text{Number of males}) \times (\text{Number of females}) $ $ \text{Total Hugs} = 6 \times 4 = 24 $ There are 24 hugs exchanged.

Calculating Total Interactions

The total number of interactions at the party is the sum of all handshakes and all hugs.

$ \text{Total Interactions} = (\text{Total Handshakes}) + (\text{Total Hugs}) $ $ \text{Total Interactions} = 21 + 24 = 45 $

Therefore, based on the conditions provided, there are a total of 45 handshakes and hugs exchanged at the tea party.