Question: There are 22 friends (A1, A2, A3....A22). If A1 has to shake hands with all of them without repeating, how many handshakes are possible?

Answer:
To determine the number of handshakes possible, we need to understand the concept of combinations.

Combinations:
In mathematics, combinations are a way to select items from a larger set without considering the order. The formula to calculate combinations is given by:

nCr = n! / (r! * (n - r)!)

Where:
- n is the total number of items in the set
- r is the number of items selected from the set
- n! represents the factorial of n (the product of all positive integers less than or equal to n)

Applying Combinations to Handshakes:
In this scenario, A1 wants to shake hands with the remaining 21 friends (A2, A3, ..., A22) without repeating any handshake.

To calculate the number of handshakes, we need to find the number of combinations of 21 items taken 2 at a time. This is because each handshake involves two people.

Using the combination formula, we can calculate the number of handshakes:

21C2 = 21! / (2! * (21 - 2)!)
= 21! / (2! * 19!)
= (21 * 20) / (2 * 1)
= 420 / 2
= 210

Therefore, the number of handshakes possible is 210.

Explanation:
By using combinations, we can calculate the number of handshakes without actually listing out all the possibilities. This saves time and effort, especially when dealing with larger sets.

In this case, we have 22 friends, and A1 wants to shake hands with each of them without repeating any handshake. By applying the combination formula, we find that there are 210 possible handshakes.

The correct answer is option B, 21.