In how many different ways can a group of twelve people be

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Thu Jul 19, 2018 8:06 pm

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In how many different ways can a group of twelve people be split into pairs ?

A 105
B 132
C 10,395
D 11,880
E 665,280

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Source: Beat The GMAT — Problem Solving | Thu Jul 19, 2018 8:06 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Thu Jul 19, 2018 10:39 pm

BTGmoderatorDC wrote:In how many different ways can a group of twelve people be split into pairs ?

A 105
B 132
C 10,395
D 11,880
E 665,280

For the first person, we can pick a pair in 11 ways;
For the second person, we can pick a pair in 9 ways (as two persons are already chosen);
For the third person, we can pick a pair in 7 ways (as four persons are already chosen);
For the fourth person, we can pick a pair in 5 ways (as six persons are already chosen);
For the fifth person, we can pick a pair in 3 ways (as eight persons are already chosen);
For the sixth person, we can pick a pair in only 1 way (as ten persons are already chosen)

So, the total number of ways = 11*9*7*5*3*1 = 10395

The correct answer: C

Hope this helps!

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Jul 20, 2018 3:27 am

BTGmoderatorDC wrote:In how many different ways can a group of twelve people be split into pairs ?

A 105
B 132
C 10,395
D 11,880
E 665,280

Since there are 12 people, the first person selected can be paired with any of the 11 other people, yielding 11 POSSIBLE PAIRS.
Since 2 of the 12 people have been selected thus far, the number of people remaining = 12-2 = 10.

Since 10 people remain, the next person selected can be paired with any of the 9 other people, yielding 9 POSSIBLE PAIRS.
Since 2 of the 10 remaining people were just selected, the number of people remaining = 10-2 = 8.

Since 8 people remain, the next person selected can be paired with any of the 7 other people, yielding 7 POSSIBLE PAIRS.
Since 2 of the 8 remaining people were just selected, the number of people remaining = 8-2 = 6.

Since 6 people remain, the next person selected can be paired with any of the 5 other people, yielding 5 POSSIBLE PAIRS.
Since 2 of the 6 remaining people were just selected, the number of people remaining = 6-2 = 4.

Since 4 people remain, the next person selected can be paired with any of the 3 other people, yielding 3 POSSIBLE PAIRS.
Since 2 of the 4 remaining people were just selected, the number of people remaining = 4-2 = 2.

The last 2 people yield 1 additional pair.

To combine the blue options above, we multiply:
11*9*7*5*3*1 = 11*63*15 = a very large integer with a units digit of 5.

The correct answer is C.

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regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

In how many different ways can a group of twelve people be

by regor60 » Fri Jul 20, 2018 6:34 am

BTGmoderatorDC wrote:In how many different ways can a group of twelve people be split into pairs ?

A 105
B 132
C 10,395
D 11,880
E 665,280

First pair can be created 12 x 11 ways. Second pair, 10X9, etc. So total ways to create pairs is 12!.

However, this gives credit to the ordering of people within a pair, which is irrelevant, so need to divide by 2^6

Similarly, 12! also gives credit to the ordering of the 6 pairs, which is also irrelevant, so need to divide by 6! also.

So, 12!/(2^6x6!) = 12x11x10x9x8x7/2^6 = 3X11x5x9x7 = c, 10395

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8136; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Jul 23, 2018 5:28 pm

BTGmoderatorDC wrote:In how many different ways can a group of twelve people be split into pairs ?

A 105
B 132
C 10,395
D 11,880
E 665,280

Twelve people can form 6 pairs.

The first pair can be formed in 12 x 11 ways, the second pair can be formed in 10 x 9 ways, and so on. Therefore, the 6 pairs can be formed 12! ways if order matters. However, since the order of the pairs does not matter, we have to divide 12! by 6!.

Furthermore, the order within each pair also does not matter, so we have to also divide by 2!. Since there are 6 pairs, we have to divide by (2!)^6 = 2^6.

Therefore, the number of ways we can split 12 people into 6 pairs is:

12!/(6! x 2^6) = (12 x 11 x 10 x 9 x 8 x 7)/2^6 = 3 x 11 x 5 x 9 x 7

We see that the a number is pretty large and its units digit must be 5, so choice C is the correct answer.

Answer: C

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