Question: 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

“A photographer will arrange 6 people of 6 different heights''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section measures the candidates’ intelligence and logical thinking abilities to solve quantitative problems. The students must answer the question by calculating it thoroughly with accurate mathematical understanding. The students must know fundamental knowledge of mathematical calculations to solve GMAT Problem Solving questions. The mathematical problems of the GMAT Quant topic in the problem-solving part is a very important part and can be solved by suitable quantitative skills.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• A photographer will arrange 6 people of 6 different heights.
• The photographer positions them in two rows of three.
• 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.
• Each person in the second row must be taller than the person standing in front of him or her.

Find Out:

• The possible number of arrangements of the 6 people.

Therefore, we can consider the arrangements as mentioned below:

4 5 6
1 2 3

2 4 6
1 3 5

2 5 6
1 3 4

3 4 6
1 2 5

3 5 6
1 2 4

Therefore the total number of arrangements of the 6 people can be possible = 5.

Correct Answer: (A)

Approach Solution 2:

The problem statement declares that:

Given:

• A photographer will arrange 6 people of 6 different heights.
• The photographer positions them in two rows of three.
• 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.
• Each person in the second row must be taller than the person standing in front of him or her.

Find Out:

• The possible number of arrangements of the 6 people.

Let’s assume that 6 people are placed in ascending order of height 1,2,3,4,5,6.

Let the positions of the six persons be

A B C
D E F

Now, it is required to fix 1 and 6 in the correct positions as per the condition of the question, therefore we can say:

_ _ 6
1 _ _

Case 1:

If Position A = 2 then, position B can only be 4 or 5, since, if B is 3, then both 4 and 5 will be greater than 3 (so, no one can take positions E and F).

Then there would not be any case possible.

2 4 6
1 3 5

or

2 5 6
1 3 4

Therefore, we get 2 solutions from this case.

Case 2:

If Position A = 3 then position B cannot be 2.

Therefore, therefore B can be 4 or 5.

3 4 6
1 2 5

or

3 5 6
1 2 4

Therefore, we get 2 solutions from this case

Case 3:

If Position A = 4, then position B can only be 5, and all other positions are also fixed.

4 5 6
1 2 3

Therefore, we get 1 solution from this case

Hence, total possible solutions are = 2 + 2 + 1 = 5
Therefore the total number of arrangements of the 6 people can be possible = 5.

Correct Answer: (A)

Approach Solution 3:

The problem statement claims that:

Given:

• A photographer will arrange 6 people of 6 different heights.
• The photographer positions them in two rows of three.
• 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.
• Each person in the second row must be taller than the person standing in front of him or her.

Find Out:

• The possible number of arrangements of the 6 people.

It is required to realise that Person1 (shortest) should be in front and Person 6 (tallest) should be in the back row.
It is also need to realize that there exists only one way of setting the three people in the front and the three people in the back.
The front row consists of "Person1, Person2 and Person5",

The people will stand like this pattern since heights of the people must lie in increasing order from left to right.

________ __________ Person6
Person1 _________ _________

Now we have only 4 people remaining (2, 3, 4, 5) and we need to select two of them for the front row. Therefore, the number of arrangements will be in 4C2 = 6 ways.

However, it is required to remember that we cannot select both taller people (Person 4 and Person 5) for the front row. This is because only then we will be left with two shorter people for the back row and Person 3 will lie behind Person 4. Remaining all cases are fine.
Therefore, possible number of ways = 6 - 1 = 5 ways

Correct Answer: (A)

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