Respuesta :
Answer:
45
Step-by-step explanation:
Let's do it for smaller numbers first.
0 people -> 0 high fives
1 people -> 0 high fives
2 people -> 1 high fives
3 people -> person 1 high fives 2 then 3
And person 2 high fives 3
->2+1= 3 high fives
4 people -> person 1 high fives 2,3, then 4
And person 2 high fives 3 then 4
And 3 high fives 4
->3+2+1=6
5 people -> person 1 high fives 2,3,4, then 5
And person 2 high fives 3,4, 5
And 3 high fives 4,5
And 4 high fives 5
-> 4+3+2+1=10
So for n people there would be (n-1)+(n-2)+....+3+2+1 high fives or (n-1)n/2 people.
Test:
3 people -> (3-1)3/2=(2)3/2=3
4 people -> (4-1)4/2=(3)4/2=6
5 people -> (5-1)5/2=(4)5/2=10
10 people-> (10-1)10/2=(9)10/2=90/2=45
or you could do 9+8+7+6+5+4+3+2+1 which also gives 45.
The other person has a great answer. Here's another approach.
Let's say we're selecting 2 people to fill slots A and B.
Slot A has 10 choices and slot B has 9 choices (since we can't pick the same person again to fill both slots simultaneously).
We have 10*9 = 90 ways to do this if order mattered.
However, we don't care about the order so we actually have 90/2 = 45 ways
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Put another way: we can form a table that has 10 rows and 10 columns.
This table has 10*10 = 100 cells inside it.
Cross off the main diagonal that runs from the upper left corner to the bottom right corner. Any of these cells represent a certain person handshaking with themself, which we don't consider.
We cross of 10 items along this diagonal so we have 100-10 = 90 cells leftover.
Now consider that person A shaking with B can be notated as AB. This is identical to BA because the order doesn't matter. So we have twice as many handshakes counted.
That's why we divide by 2 to go from 90 to 90/2 = 45
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Another way to find the answer is to use the nCr combination formula with n = 10 and r = 2.
Yet another way to find the answer is look at Pascal's triangle. Find the row that has 1,10,45... in it at the start. These values correspond to r = 0, 1 and 2 respectively. We can see that 45 is when r = 2, which means we have 2 people shaking hands and there are 45 ways to have ten people shake hands.
So there are many approaches you can take with this problem.
