SOLUTION: There are five people in a room. Each person shakes the hand of every other person exactly one. How many hands shakes are exchanged

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Question 1152008: There are five people in a room. Each person shakes the hand of every other person exactly one. How many hands shakes are exchanged

Found 3 solutions by MathLover1, greenestamps, ikleyn:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Well when people meet each other the total number of handshakes that take place would always be the total sum of numbers from to
That means if people meet the total number of handshakes be

so, hands shakes are exchanged
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Here is a more sophisticated method for counting the number of handshakes; it is especially useful when the number of people is large.

Suppose the number of people is n. Then each of the n people shakes hands with each of the other (n-1) people, making a total of n*(n-1) handshakes.

But when you count all of the handshakes by all the people in that way, you count each distinct handshake twice; so the actual number of handshakes is (n*(n-1))/2.

In this problem, with 5 people, that gives you (5*(5-1))/2 = (5*4)/2 = 10.

Of course, adding 4+3+2+1 is faster than that.... But if the number of people were 79 instead of 4, you wouldn't want to be adding 78+77+76+...+3+2+1 to find the number of handshakes.
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.

            Standard (but not unique (!) ) solution to this problem is as follows. 


Each person makes  (n-1) handshakes, so your first intention is to answer n*(n-1) handshakes, in all.


But be careful (!)


By counting in this way, we count each handshake twice, for every of the two hand shakers. 


Therefore, we must divide  n*(n-1)  by 2, and then we come to the correct answer:


    the number of handshakes is  .    ANSWER

Solved.

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It is classic problem (!)

MEMORIZE its solution (!)



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