there are 12 people at a party. each person shakes hands with each of th other guests. how many hadshakes will there be in all
each of the 12 shakes hands with 11 others, but A shaking with B is the same as B shaking with A, so
12 x 11/2 = 66
In combination notation:
C(12,2) = 12!/(10!2!) = 66
Another viewpoint:
A shakes the hand of 11 people.
B shakes the hand of 10 people, already having shaken the hand of A.
C shakes the hand of 9 people, already having shaken the hand of A and B.
Therefore, the total number of hand shakes is simply the sum of the numbers from 1 through 11 or, S = n(n + 1)/2 = 11(12)/2 = 66.
1,993,373
You might want to recheck the math. If you start with #12 that's 11 shakes then 10, 9,8,7,6,5,4,3,2. when you get to #2 he shakes with #1 that is 65 #1 has shook with everybody so doesn't need to shake with anyone.
To calculate the total number of handshakes at the party, we can use a combination formula.
In this case, there are 12 people at the party, and each person shakes hands with every other guest. To find the number of handshakes, we need to calculate the number of combinations of 2 guests that can be formed from a group of 12 people.
The formula for combinations is given by:
C(n, r) = n! / (r! * (n-r)!)
Where:
n is the total number of items in a set (12 in this case)
r is the number of items chosen from the set to form a combination (2 in this case)
! denotes the factorial of a number, which means multiplying all the whole numbers from 1 to that number.
Plugging in the values, we get:
C(12, 2) = 12! / (2! * (12-2)!)
= 12! / (2! * 10!)
Calculating the factorials:
12! = 12 * 11 * 10!
2! = 2
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Now we can substitute these factorials into the formula:
C(12, 2) = (12 * 11 * 10!) / (2 * 10!)
The 10! terms cancel out:
C(12, 2) = (12 * 11) / 2
Calculate the multiplication:
C(12, 2) = 132 / 2
Finally, divide to get the answer:
C(12, 2) = 66
Therefore, there will be a total of 66 handshakes at the party.