A photographer will arrange 6 people of 6 different heights

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

Time for the OA.

OA: A (5)

The key to solving this problem is recognizing that the positions of the tallest (6) and shortest (1) people are fixed: the tallest must be in the second row on the right, and the shortest in the first row on the left:

x x 6
1 x x

Once these positions are set, the arrangement of the remaining four people leads to five possible configurations. Thus, the correct answer is (A) 5.

Let me try,

The arrangement will be something like this.

_ _ _
_ _ _

Position of A and F has to be fix i.t. bottom line left for A and upper line right for F. Other position needs to be filled-in by B, C, D and E.

B can be on right or A or behind A. So 2 positions possible. C can be on right of B or behind him or A, that means 3 possible positions. D and E will have to fit into the positions accordingly without any option. Hence the ans should be 2 X 3 = 6.

Is that right?

Position of 1 and 6 is fixed. Still don't see a formula for the rest, except manually finding the arrangement, which as you said takes more than 2 mins!

I got 6 as answer too, manually arranging assuming 1,2,3,4,5,6 as heights (since each is different from other)

so possible positions are as below:

456 436
123 125

346 356
125 124

246 256
135 134

Kalpesh logic looks right to me... and is faster than listing possibilities

u missed
436
125
[/quote]

But 436 wouldn't be correct as from left to right it has to in increasing order and 4 is greater than 3.
Unless I'm not getting it?[/quote]


u r right, i got
436
125
incorrectly...

so the answer is 5 ?

Solution ( courtesy Manhattan GMAT staff)

BL BM BR
FL FM FR

("B" stands for "back", "F" stands for "front", "L" stands for "left", etc.)

Let's also assign "names" to each of the six people - 1 is the shortest, 2 is the next shortest, ... and 6 is the tallest.

Notice first that the only place where 6 can stand is in the BR position. A person standing in any of the other positions has to be shorter than at least one other person, and 6 isn't shorter than anybody.

By similar reasoning, we can see that the only place where 1 can stand is in the FL position. A person standing in any of the other positions has to be taller than at least one other person, and 1 isn't taller than anybody.

So we know that any possible arrangement will be of this form:

BL BM 6
1 FM FR

All we need to do is count possible ways of putting 2, 3, 4, and 5 in positions BL, BM, FM, and FR. In order to count possibilities, let's focus on who goes into the BL position. 1 and 6 are already fixed in their own positions. There's no way 5 could be in the BL position, because there would be no way to assign someone to BM such that the heights in the back row increased consistently from left to right. So we know that the person in the BL position has to be either 2, 3, or 4. We investigate each possibility in turn:

If 2 goes in the BL position, there are just two possibilities:

2 4 6
1 3 5

and

2 5 6
1 3 4

If 3 goes in the BL position, there are also two possibilities:

3 4 6
1 2 5

and

3 5 6
1 2 4

If 4 goes in the BL position, there is just one possible arrangement:

4 5 6
1 2 3

Counting these possibilities, we see that there are only 5 possible arrangements.

I immediately figured out that Position 1 and 6 were fixed, but then I ended up wasting my time trying to find a formula for the whole thing. After a while I stopped trying to find a formula and just counted the possible combinations and came up with 5.

Here is a way to do this question without using brute force -

Assume the 6 people in ascending order of height are 1,2,3,4,5,6.

Let the positions be

A B C
D E F

Now, it is obvious that we have to fix 1 and 6 in the correct positions as shown below:

_ _ 6
1 _ _

Case 1:

Position A = 2

In that case, position B can only be 4 or 5, because, if B is 3, then both 4 and 5 will be greater than 3 (so, none can take positions E and F), and there won't be any case possible.

2 4 6
1 3 5

or

2 5 6
1 3 4

So, 2 solutions.

Case 2:

Position A = 3

In that case, position B cannot be 2. hence only 4 and 5 can take its place.

3 4 6
1 2 5

or

3 5 6
1 2 4

So, 2 solutions.

Case 3:

Position A = 4

In that case, position B can only be 5, and all other positions are also fixed.

4 5 6
1 2 3

So, 1 solution.

Hence, total solutions = 2 + 2 + 1 = 5

This can be done quite fast, but for the purpose of explaining, I have tried to be more elaborate.

Hi All,

In Quant questions such as these, the number of possible arrangements is sometimes so limited that you can actually just list them out - in that way, you can visualize the solution and just do a bit of 'brute force' work:

Here, we're told to arrange 6 people (who all have DIFFERENT heights) into two rows of 3 so that each person in the first row is 'in front' of a taller person in the second row AND heights increase from 'left to right.' We're asked how many arrangements of people are possible.

If we label the six people as 1, 2, 3, 4, 5, 6 (with 1 the shortest, and each number being 'taller' than the one immediately preceding it), we would have the following options...

456
123

356
124

346
125

256
134

246
135

There are NO other options.

Final Answer:

GMAT assassins aren't born, they're made,
Rich

I'm getting 15. Please see the attachment.

Hi kham71,

A number of your examples do NOT follow the 'restrictions' that were mentioned in the prompt. We're told that...

"The heights of the people within each row must INCREASE from left to right...."

So, while 123 would be allowed, 213 would NOT. Keeping that restriction in mind, how many of your possibilities can you eliminate?

GMAT assassins aren't born, they're made,
Rich

I got this problem wrong when I encountered it during my practice test but looking at the solution and applying some reverse thinking I came up with following explanation which makes it little easier to quickly solve the problem.

Let's say we have 6 heights = {1, 2, 3, 4, 5, 6}

1st arrangement is obvious which is as following:

4 5 6 <- second row
1 2 3 <- first row

Now we need to think about other possible arrangements with given restrictions.

Couple of things we need to note:

1) Since 1 is smallest and 6 is greatest, they CANNOT be placed elsewhere from their current position. Hence their position is fixed and we need to only concern ourselves with position of rest of the 4 numbers i.e. {2, 3, 4, 5}

2) For other possible arrangements, the numbers from first row will have to go into second and from second row to 1 first row.

3) From {2, 3, 4, 5}, you cannot move both {2, 3} to second row (with {4, 5} moved to 1st row) as it will not satisfy the constraints in any arrangement.

4) Now looking at numbers just from second row, let's say if either of them has to placed in 1st row, they can only occupy 3rd position. Hence we have following two case:

Case 1: Placing {4} in third position of the first row, find arrangements of {2, 3, 5} that satisfy the constraints given in the problem.

_ _ 6
1 _ 4

Case 2: Placing {5} in third position of the first row, find arrangements of {2, 3, 4} that satisfy the constraints given in the problem.

_ _ 6
1 _ 5

In each of the two cases above, you will find two arrangements that satisfy constraints.

Case 1:
1st arrangement:
3 5 6
1 2 4
2nd arrangement:
2 5 6
1 3 4

Case 2:
1st arrangement:
3 4 6
1 2 5
2nd arrangement:
2 4 6
1 3 5


Hence, total arrangements: 1 + 2 + 2 = 5 arrangements.

Hi Folks,

Just wanted to share my trick to solve this problem.

To answer this choice, you need to know that the positions of 1 and 6 are fixed because they are the lowest and highest.
x y 6
1 v w

v, w, x and y can be any number that create possibility
Therefore, we need to see check the possibility by trial.
The trick here is only v and w that are in play here --> x and y will be automatically fitted from the rule.

Case 1 v,w = 2 and 3
_ _ 6
1 2 3

we got
4 5 6
1 2 3

Case II v,w = 2 and 4
_ _ 6
1 2 4

we got
3 5 6
1 2 4

Case III v,w = 2 and 5
_ _ 6
1 2 5

we got
3 4 6
1 2 5

Case IV v,w = 3 and 4
_ _ 6
1 3 4

we got
2 5 6
1 3 4

Case V v,w = 3 and 5
_ _ 6
1 3 5

we got
2 4 6
1 3 5

Case VI v,w = 4 and 5
_ _ 6
1 4 5

we are left with 2 and 3 which both are less than 4

So, Case VI is no longer valid
Ultimately, we got 5 cases.

Hope this help

Cheers,
Katanyu

But can't P4 and P5 be arranged in two ways in the front, as P4.P5 or P5.P4. Then, by this method the answer would be (6- 2) = 4 which is incorrect.

What am I missing?

---> stupid qn on my part...can't be P5.P4 from the left because "The heights of the people within each row must increase from left to right"

4C2 = 6 is the number of ways of selecting 2 people out of 4.
So out of P2, P3, P4 and P5, you can SELECT 2 in 6 ways - (P2, P3), (P2, P4), (P2, P5), (P3, P4), (P3, P5), (P4, P5)

P4, P5 will appear only once in 6.
P4, P5 and P5, P4 is an arrangement but we are only selecting.

We can let the 6 people be A, B, C, D, E, and F, with heights a, b, c, d, e, and f, and a < b < c < d < e < f.

Thus we can have an arrangement that is:

1st row: A, B, C
2nd row: D, E, F

Notice that since A is the shortest, he must be the first (or leftmost) one in the first row. Similarly, since F is the tallest, he must be the last (or the rightmost) one in the second row. Thus the question becomes how many ways to arrange B, C, D and E.

We also see that B can stand in one of the following two positions: either second in the first row or first in the second row. If B is second in the first row, then the arrangements could be either the one listed above or the two listed below:

1st row: A, B, D
2nd row: C, E, F

1st row: A, B, E
2nd row: C, D, F

If B is first in the second row, then the arrangements could be:

1st row: A, C, D
2nd row: B, E, F

1st row: A, C, E
2nd row: B, D, F

Since B can’t stand anywhere else, we have listed all the possible arrangements. We see that there are 5 such arrangements.

Answer: A

Arrangements possible:-

123
456

135
246

125
346

124
356

134
256

IMO A