14 students in an Algebra class were asked to shake hands with each other. Each student shook hands with each of the other students exactly once. How many handshakes occured in this class?
Each of the 14 student shakes hands with the 13 remaining students for a total of 13*14 times. However, each handshake was counted twice, once from each student, so the above count has to be divided by 2 to get:
13*14/2=91 handshakes.
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To find the number of handshakes that occurred in the Algebra class, we need to use the concept of combinations.
First, let's understand the problem. There are 14 students in the class, and each student shakes hands with all the other students exactly once. So, we need to calculate how many ways we can choose two students out of 14 to shake hands.
Now, to calculate the number of handshakes, we can use the combination formula:
C(n, r) = n! / (r! * (n - r)!)
Where:
n is the total number of students (14 in this case)
r is the number of students selected to shake hands at a time (2 in this case)
! denotes the factorial of a number, which is the product of all positive integers less than or equal to that number
Let's substitute the values into the combination formula:
C(14, 2) = 14! / (2! * (14 - 2)!)
Now, let's simplify the equation:
14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
2! = 2
(14 - 2)! = 12!
C(14, 2) = (14 * 13 * 12!) / (2 * 1 * 12!)
Now, we can cancel out the common term 12!:
C(14, 2) = (14 * 13) / (2 * 1)
C(14, 2) = 91
Therefore, there were 91 handshakes that occurred in the Algebra class.