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

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Joined: 10 Feb 2011
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If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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09 Mar 2011, 15:12
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Difficulty:

15% (low)

Question Stats:

68% (00:43) correct 32% (00:48) wrong based on 561 sessions

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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

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Joined: 02 Sep 2009
Posts: 59756
Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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09 Mar 2011, 15:20
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banksy wrote:
. 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

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

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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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29 Apr 2014, 12:17
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We got #10 people who shake each other's hands once ==> a pair of 2

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

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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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16 Sep 2015, 21:03
Handshakes are made in a pair of 2
Hence the total handshakes = total different different pairs

This can be found by 10C2= 10!/8!*2!

On solving , we get 45 (Option D)
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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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20 Mar 2018, 11:46
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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.

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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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08 Sep 2019, 10:07
banksy wrote:
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

Project PS Butler : Question #98

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total handshakes ; 10c2 ; 45
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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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12 Nov 2019, 20:32
1
banksy wrote:
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

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.

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Re: If 10 persons meet at a reunion and each person shakes hands exactly  [#permalink]

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16 Nov 2019, 16:32
banksy wrote:
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

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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
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Re: If 10 persons meet at a reunion and each person shakes hands exactly   [#permalink] 16 Nov 2019, 16:32
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