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Q: How many hand shakes would take place if 20 people shake hands once?

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72

For everyone to shake hands with everyone else, there are 21 handshakes. (below copied from my answer of a similar question) This is an arithmetic progression and can be solved with the equation Sn=(1+(n-1))(n-1)/2 where Sn is the total sum of handshakes for n people. NB: I have used n-1 instead of n in the equation for the sum of an arithmetic progression, because you're not really going to shake hands with yourself, so you don't include the nth term, in this case 7. Alternatively, you can solve this geometrically by drawing a heptagon, drawing lines between all the vertices, and then adding all the lines up.

10 times

The first person can be selected in one of seven ways. Having selected him, the second can be selected from the remaining six in six ways. So there would appear to be 7*6 ways of selecting the couples shaking hands. But, x shaking y's hand is the same handshake as y shaking x's hand. Thus each handshake is conted twice, So the total number of handshakes is 7*6/2 = 21

You can shake the bed with me;)

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25 shakes

36. Everybody shakes 8 hands but each shake counts for 2 people. So 9*8/2=36.

45 handshakes

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20 hand shakes would take place. Here's how: 1234567 23456 34567 4567 567 67 7 And a person can't shake his own hand or someones hands twice

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72

If each person is shaking with only one hand, then the answer is seven. If they are shaking with both hands, then the answer is 14.

105 ( First person shakes 14 different hands, second shakes 13 etc etc down to 14th shakes 1 hand. Sum of 1 to 14 = 105.)

the first shakes 8 people's hands (remember, not his own), the second 7 (he doesn't shake the first one's hand), then the third shakes six, the fourth shakes 5, the fifth shakes 4, the sixth shakes 3, the seventh shakes 2, and the 8th shakes the 9ths hand so 8+7+6+5+4+3+2+1 = 36

one....

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