105 hand-shakes

Expert replies

sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Wed Feb 11, 2009 5:02 am

Persons gathered in a meeting shook hands once among each other, with no one to spare. An un-invited was happy counting the total number of hand-shakes that took place that way, and finally he murmered: "Wow! 105 hand-shakes!" How many persons were attending the meeting there?
A. 105
B. 35
C. 25
D. 15
E. 12

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Source: Beat The GMAT — Problem Solving | Wed Feb 11, 2009 5:02 am

mals24: Legendary Member; Posts: 683; Joined: Tue Jul 22, 2008 1:58 pmFollowed by: 2 members; Location: Dubai; Thanked: 73 times

by mals24 » Wed Feb 11, 2009 5:30 am

I guess D

Let the total number of guests = N

Total number of handshakes = N(N-1)/2 = 105

N(N-1) = 210

N = 15

sureshbala: Master | Next Rank: 500 Posts; Posts: 319; Joined: Wed Feb 04, 2009 10:32 amFollowed by: 9 members; Location: Delhi; Thanked: 84 times

by sureshbala » Wed Feb 11, 2009 5:40 am

If there are n persons in the party, since each person shakes hand with all the others exactly once, the total number of shake hands will be nC2.

Given nC2 = 105. Hence n = 15

ellexay: Senior | Next Rank: 100 Posts; Posts: 57; Joined: Sat Jan 24, 2009 7:45 am; Thanked: 1 times

by ellexay » Wed Feb 11, 2009 5:41 pm

Suresh,

How did you find out the value of N in NC2=105? Is there a formula?

Thanks!

DeepakR: Master | Next Rank: 500 Posts; Posts: 158; Joined: Tue Sep 30, 2008 8:47 amFollowed by: 1 member; Thanked: 23 times; GMAT Score:660

by DeepakR » Wed Feb 11, 2009 5:53 pm

Ncr=n!/(n-r)!*r! so Nc2=n!/(n-2)!*2!=n*(n-1)*(n-2)!/(n-2)!*2!
hence n*(n-1)/2=210 solving we get n=15

- Deepak

ven4gmat: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Sun Feb 08, 2009 11:29 am; Thanked: 3 times

by ven4gmat » Thu Feb 12, 2009 1:48 am

If n is the number people in the party, then the total number of shakehands will be nC2.

given nC2 = 105

i.e n(n-1)/2 = 105.

So n =15

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