6
Let the four people each shake hands with each of the others once as A, B, C, and D.
The illustration of the event is as follows:
We can observe that there are 6 handshakes among four people in a room.
The order is not noticed.
Calculation:
A combination is used to calculate how many ways to choose or know the various arrangements by not considering the order.
The formula for finding the number of different ways or number of combinations of n different objects taken r at the time is
[tex]\boxed{\boxed{ \ _nC_r \ or \ C(n, r) = \frac{n!}{r!(n - r)!} \ }}[/tex]
Given:
Let us evaluate the value of ₄C₂.
[tex]\boxed{ \ _{4}C_2 \ or \ C(4, 2) = \frac{4!}{2! \cdot (4 - 2)!} \ }[/tex]
[tex]\boxed{ \ _{4}C_2 = \frac{4!}{2! \cdot 2!} \ }[/tex]
Recall [tex]\boxed{n! = n \times (n-1) \times (n-2) \times ... \times 3 \times 2 \times 1}[/tex] as n factorial.
[tex]\boxed{ \ _{4}C_2 = \frac{4 \times 3 \times 2!}{(2 \times 1) \cdot 2!} \ }[/tex]
We expand 4! because there are 2! inside it. Then we easily cross out 2! in the numerator and denominator.
[tex]\boxed{ \ _{4}C_2 = \frac{4 \times 3}{2} \ }[/tex]
[tex]\boxed{ \ _{4}C_2 \ or \ C(4, 2) = \frac{12}{2} \ }[/tex]
As a result, the expression ₄C₂ is 6.
Thus, the number of handshakes that occur is 6.
Keywords: four people in a room, each shakes hands, how many, a combination, many ways, the various arrangements, by not considering the order, the formula, finding the number of different ways, n different objects taken at that time, factorial
