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number of arrangements

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number of arrangements

by daretodream » Fri Feb 19, 2010 3:28 am
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 someone in the second row. The heights of the people within each row must increase from left to 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
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by harsh.champ » Fri Feb 19, 2010 3:52 am
daretodream wrote: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 someone in the second row. The heights of the people within each row must increase from left to 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
Now, let the heights in ascending order be P1,P2---P6.

So, if P1 has to be on the 1st row left.
Also,P6 has to be on the 2nd row right .
So, basically we have to see the arrangement of P2,P3,P4,P5.
Case 1:-P2 in 1st row middle.
The selection of 1 person to be on the right of P2 can be done in 3C1 ways.

Case 2:-If P3 in 1st row middle.
In this case,P2 has to be above P1 (i.e. 2nd row left)
Thus in 2! ways P4 and P5 can exchange their positions.

Thus,answer would be [spoiler]5.(A)[/spoiler]
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by vijay_venky » Fri Feb 19, 2010 7:54 pm
Just one more way of doing this,

1. Let the heights be h{1,2,3,4,5,6}
2. h1 should be in the first row.
3. h6 should be in the second row.
4. So the positions occupied by them are fixed.

|_|_|6|
|1|_|_|

Now we need to find out the ways in which we can arrange others.
In the second row (the row with 1), the possible combinations are
[spoiler]23
24 34
25 35 [/spoiler]

For each of this, there is only one way of arrangement because of the rule of ascending order.

So only 5 possibilities of the arrangement
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by shashank.ism » Sat Feb 20, 2010 1:18 am
I would say its a very nice question... though I have some confusion with the answer, though I ma trying to sort it out...its nice that venky and harsh has posted ans of such tricky ques in nice manner....thanks...
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