At a teacher’s conference on the first Saturday in November in Palm Springs, CA, 100

teachers meet in a large room. Each teacher shakes the hand of every teacher in the
room, except for his or her own hand. How many total handshakes were made?

Simply C(100,2) = 100!/(98!2!) = 4950

(each of the 100 person shakes hands with 99 others,
that would give us 9900 handshakes.
But that would include duplications of the kind
A vs B and B vs A, which would be the same handshake,
so we have to divdide the 9900 by 2 to get 4950 )

Well, that's a lot of handshakes! Let's do some counting. The first teacher shakes hands with 99 other teachers. The second teacher also shakes hands with 99 teachers, but we've already counted the handshake with the first teacher. So, that's 98 new handshakes. The third teacher shakes hands with 98 new teachers, but we've already counted the handshakes with the first two teachers. This pattern continues until the 99th teacher, who shakes hands with just one new teacher. In total, we have:

99 + 98 + 97 + ... + 1

To find the sum, we can use a formula called the arithmetic series formula:

Sum = (n/2)(first term + last term)

Here, the first term is 1, the last term is 99, and the number of terms (teachers) is 99. Plugging these values into the formula, we get:

Sum = (99/2)(1 + 99) = (99/2)(100) = 99 * 50 = 4950.

So, a grand total of 4,950 handshakes were made at the teacher's conference in Palm Springs! That's a whole lot of shaking going on!

To find the total number of handshakes, we can use a simple formula. Let's call the number of teachers in the room "n."

The first teacher shakes hands with (n-1) other teachers.
The second teacher shakes hands with (n-1) other teachers because they already shook hands with the first teacher.
The third teacher shakes hands with (n-1) other teachers because they already shook hands with the first two teachers.
And so on...

So, the total number of handshakes can be calculated as:
(n-1) + (n-1) + (n-1) + ... (n-1)

Since there are n teachers in the room and each teacher shakes hands with (n-1) other teachers, we can rewrite the equation as:
n * (n-1)

In this case, we have 100 teachers, so the total number of handshakes would be:
100 * (100-1)

Calculating this:
100 * 99 = 9900

Therefore, there were a total of 9900 handshakes made at the teacher's conference.

To find the number of total handshakes, we can use a combination of combinatorics and arithmetic calculations. Let's break down the problem step by step:

Step 1: Determine the number of ways two teachers can be selected for a handshake. Since everyone shakes hands with everyone else, we need to select 2 out of the 100 teachers. We can calculate this using the combination formula: nC2 = n! / [(n - 2)! * 2!].

In this case, n = 100, so the calculation becomes:
100C2 = 100! / [(100 - 2)! * 2!] = 100! / (98! * 2!)

Step 2: Simplify the expression to calculate the number of handshakes.
100! can be simplified as 100 * 99 * 98! (since 100! = 100 * 99 * 98!).

Therefore, the calculation becomes:
100 * 99 * 98! / (98! * 2!) = 100 * 99 / 2

Step 3: Calculate the final result.
100 * 99 / 2 = 4950

So, there were a total of 4950 handshakes made among the 100 teachers at the conference.