Answer:
Explanation:
When 8 people shake hands with one another exactly once, you can calculate the total number of handshakes using the formula for combinations (also known as "n choose k" or binomial coefficients). The formula for combinations is:
C(n, k) = n! / (k! * (n - k)!)
In this case, n represents the number of people (8), and k represents the number of people you choose to shake hands with at any given time (2, since each handshake involves two people). So, you want to calculate C(8, 2):
C(8, 2) = 8! / (2! * (8 - 2)!)
Now, calculate each factorial:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
2! = 2 * 1 = 2
(8 - 2)! = 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720
Now, plug these values into the combinations formula:
C(8, 2) = 40,320 / (2 * 720) = 40,320 / 1,440 = 28
Answer: So, there are a total of 28 handshakes that take place among the 8 people when each person shakes hands with everyone else exactly once.
Generalised formula
To generalize the formula for calculating the number of handshakes among n people, you can use the formula for combinations (n choose 2), where n represents the total number of people. The formula is:
C(n, 2) = n! / (2! * (n - 2)!)
In this formula:
n is the total number of people.
2 represents the number of people involved in each handshake.
C(n, 2) calculates the number of unique pairs of people who can shake hands with each other exactly once.
You can use this formula to find the number of handshakes for any value of n, where n is the total number of people.