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No. of ways - P&C

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No. of ways - P&C

by puneetpjl » Sun Jul 29, 2012 11:13 pm
8 women of 8 different heights are to pose for a photo in 2 rows of four. Each woman in the second row must stand directly behind a shorter woman in the first row. In addition, all of the women in each row must be arranged in order of increasing height from left to right. With these restrictions fully adhered to, in how many different ways can the women pose?

A) 2
B) 14
C) 15
D) 16
E) 18
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by armand_h » Mon Jul 30, 2012 8:15 am
Let's replace the women with numbers from 1 to 8

for example:

|5|6|7|8|
|1|2|3|4|

to compute the number of combinations with increasing left to right and bottom to top, you need to fill the table below from 1 to 8 starting from left to right and bottom to top:
|x|x|x|x|
|x|x|x|x|

for example:
|3|x|x|x|
|1|2|4|x|
you should not leave an empty square on the bottom or to the left, for example you should not have something like:
|x|x|3|x|
|1|2|x|x|

Which means that for each combination at the bottom line there is one and only one combination for the top line. So the total number of combinations is equal to the number of combinations of the bottom line.

Here is a tree that builds all the combinations for the first line.
the correct answer is 14 combinations
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by puneetpjl » Tue Jul 31, 2012 1:30 am
No Armand. still didnt get it. i am unable to understand the relation. i solved the question by listing all the possibilities but need help with this relation.
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by GMATGuruNY » Tue Jul 31, 2012 3:38 am
The question posted above would never appear on the GMAT.
The following version is more viable:
A photographer will arrange 6 people of 6 different heights for photograph by placing them in two rows of three so that each person in the first row is standing in front of some one in second row. The heights of the people with in each row should be increasing from left right, and each person in the second row must be taller than the person standing in front of him or her. How many such arrangements of the 6 people are possible?

a) 5
b) 6
c) 9
d) 24
e) 36
Because here there are only 6 people, the number of possible arrangements can be counted relatively quickly.

Let the 6 people be represented by the numbers 1-6, inclusive.
Fill the MOST RESTRICTED positions first and work down to the LEAST RESTRICTED positions.

Place 1 and 6:
XX6
1XX

Place 2:
Case A:
2X6
1XX

Case B:
XX6
12X

Place 5, whose position will determine where 3 and 4 can go:
Case A:
256...246
134...135

Case B:
346...356...456
125...124...123

Total options = 5.
The correct answer is A.
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by rideema » Thu Aug 02, 2012 3:06 am
Thank u Armand. Got it !! Pretty good one puneetpjl.. :)
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