If 10 persons meet at a reunion and each person shakes hands exactly

Question

If 10 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes?

(A) 10•9•8•7•6•5•4•3•2•1
(B) 10•10
(C) 10•9
(D) 45
(E) 36

Bunuel · Accepted Answer

The total number of handshakes will be equal to the number of different pairs possible from these 10 people (one handshake per pair), so  C^2_{10}=45 .

Answer: D.

EMPOWERgmatRichC · Answer

Hi All,

This question can be solved in a couple of ways: a high-concept math approach or a "brute-force" answer that anyone can use. I'll focus on the second method.

Since we have 10 people, who will all shake hands with one another, we know that each pair of people will lead to 1 hand shake (and a person CAN'T shake hands with himself or herself).

If we call the people ABCDE FGHIJ

Person A will shake hands with BCDE FGHIJ = 9 shakes

Person B ALREADY shook hands with A, so they won't shake hands again…. 
Person B will shake hands with CDE FGHIJ = 8 shakes

Person C ALREADY shook hands with A and B, so they won't shake hands again…. 
Person C will shake hands with DE FGHIJ = 7 shakes

Notice the pattern 9, 8, 7…..the numbers will shrink by 1 with every letter, so we'll end up with…

9+8+7+6+5+4+3+2+1+0 = 45 total handshakes.

Final Answer:  D

GMAT assassins aren't born, they're made, 
Rich

CarloCjm · Answer

We got #10 people who shake each other's hands once ==> a pair of 2

10!/8!2! = 10*9 / 2*1 = 45.

Hence answer D.

ScottTargetTestPrep · Answer

The total number of handshakes is 10C2 = (10 x 9)/2! = 45.

Alternate Solution:

Let’s call the 10 individuals A, B, C, D, E, F, G, H, I , and J.

A will shake hands with 9 others (since he doesn’t shake hands with himself). B will shake hands with 8 others (excluding himself and A, with whom he has already shaken hands). C will shake hands with 7 others (excluding himself and A and B). This pattern continues, with each person shaking hands with 1 fewer individual, resulting in the sum 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0 = 45.

Answer: D

TheNightKing · Answer

Let's say every person shakes hands with every other person. For one person, there will be 9 handshakes so for 10 it will be 90
But we counted twice. A with B and B with A is same thing. So we need to 90/2=45

einstein801 · Answer

Why does 10C2 represents the number of different pairs of handshakes?

Bunuel · Answer

10C2 represents the number of pairs possible from these 10 people. Since there is one handshake per pair, then the total number of handshakes is also 10C2.

bumpbot · Answer

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If 10 persons meet at a reunion and each person shakes hands exactly

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If 10 persons meet at a reunion and each person shakes hands exactly

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