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How to solve Complex Arrangement Problems using Tables and Matrix

Sunday, June 14th, 2020

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Arrangement problems are an integral part of all the curriculum of Logical Reasoning exams. These are the most common exam and you can find them all kind of entrance exams involving aptitude test such as CAT, XAT, SBI PO etc. This signifies the importance of the understanding of these kinds of problems. There are various kinds of arrangement problems linear, circular and table or matrix based complex arrangement problem. And, today We’ll explain these tabular or matrix based complex arrangement problems.

Complex arrangement problems are very much similar to linear arrangement problem except for some extra dimensions in them.

Following example will:

There are five villages, A, B, C, D and E. Two of these have a post office, three have schools and three are accessible by road. Two have a population of more than 5000 each, two have population between 2000 and 5000 each, and one has a population of less than 2000. Two of these villages have electricity in addition to certain other facilities such as schools and accessibility by road. One village with population of less than 2000 also has a school but neither has a post office nor is accessible by road while each village with population of more than 5000 has a school. Of the two the village having population between 2000 and 5000, only one has a post office and is accessible by road. A is accessible by road. B has a population of more than 5000, D has a school and is accessible by road but does not have a post office while E has a school but is not accessible by road.

As you can notice through the above illustration that rather than one characteristic feature it has 3, unlike linear arrangement problems which have straightforward information. These extra features make these problems complex and not the solution. The solution to these problems is quite simple and there are two methods to acquire it. These are

• Matrix Method
• Table Method

I know you might be wondering that what the difference is between the two. No worries! I will vividly describe the application of both of these methods along with an example.  Let’s first start with the Matrix Method.

In this method, a table is drawn and the first column is reserved for the objects or people that whose properties are given in the problem. The other columns are used to list those various features associated with the object. A prototype is given below.

	Post Office	School	Accessibility by road	Electricity	Population
					Less than 2000	Between 2000 and 5000	Greater than  5000
A
B
C
D
E

Now the next step is to start reading putting up the direct information associated with the object given in the problem by using ‘+’ for yes and * for no. Let’s start doing it for the above given information.

It’s been given directly that

• A is accessible by road.
• B has a population of more than 5000.
• D has a school and is accessible by road but does not have a post office.
• E has a school but is not accessible by road.

Now put + and * signs in all the required places like this.

	Post Office	School	Accessibility by road	Electricity	Population
					Less than 2000	Between 2000 and 5000	Greater than  5000
A			+
B					*	*	+
C
D	*	+	+
E		+	*

Now using the above bold points we need to estimate the link with what other information is given in the problem that relates them with other characteristics.

It’s been given that the village with population more than 5000 has a school which implies B has a school and this point also has another implication that there are total three schools and we just established on in B and other two are in D and E which means either D or E has a population greater than 5000. To figure out which one has the population we can use another fact provided above i.e. one village with the population of less than 2000 also has a school but neither has a post office nor is accessible by road. Now, since D is accessible by road, therefore, it can’t be the one with the population less than 2000 and hence, E is the one with population less than 2000. Also, A and C does not have schools. Now let’s add in this information in our matrix.

	Post Office	School	Accessibility by road	Electricity	Population
					Less than 2000	Between 2000 and 5000	Greater than  5000
A		*	+		*		*
B		+			*	*	+
C		*			*		*
D	*	+	+		*	*	+
E	*	+	*		+	*	*

As you can see we have almost covered more than half of the matrix and we can clearly find that both A and C are the villages have the population between 2000 & 5000. And only one of them has a post office and accessibility by road. Since A is accessible by road, therefore, it must have a post office. Also, three villages have road accessibility we have established two of them and one is left which should be B as only one the village having population between 2000 and 5000, only one has a post office and is accessible by road thus C can’t be the one and thus also implies that B has the post office as well. Also, two of these villages have electricity in addition to certain other facilities such as schools and accessibility by road and the only suited options that fulfill these criteria is B and D. Hence, the required matrix is:

	Post Office	School	Accessibility by road	Electricity	Population
					Less than 2000	Between 2000 and 5000	Greater than  5000
A	+	*	+	*	*	+	*
B	+	+	+	+	*	*	+
C	*	*	*	*	*	+	*
D	*	+	+	+	*	*	+
E	*	+	*	*	+	*	*

Now the matrix is completed and in the exam you will be provided some questions like the ones that are given below and you have to answer them on the basis of this matrix.

Complex Arrangement Example 1:

Which village has a post office along with a school?

a.) A

b.) B

c.) C

d.) D

e.) E

Complex Arrangement Example 2:

Which village does not have any of the facilities available to the other villages?

a.) A

b.) B

c.) C

d.) D

e.) E

Complex Arrangement Example 3:

Which two have electricity?

a.) A,B

b.) B,C

c.) B,D

d.) C,D

e.) None of the above

Complex Arrangement Example 4:

Which three villages are accessible by road?

a.) A,B,D

b.) A,B,E

c.) A,D,E

d.) B,D,E

e.) A,B,C

We can now move forth to the second method the Table Method and we will again use an example to understand the use of this method.

A, B, C, D, E, F and G are seven friends studying seven different branches of engineering, namely Mechanical, Chemical, Electrical, Electronics, Civil, Computer and Aeronautical Engineering, not necessarily in this order. Each of them studies in three different colleges, X, Y and Z. Not less than two study in any college. D studies Electrical engineering in College X. The one who studies Chemical Engineering does not study in college Z. F studies Aeronautical Engineering in college Y with only B. A does not study in college X and does not study Civil Engineering. E studies Computer Engineering and does not study in college X. G studies Electronics Engineering but not in college X. none in college X studies Mechanical or Civil Engineering.

Complex Arrangement Example 1:

Which of the following groups represents the persons studying in college Z?

a.) D, B

b.) C, E, G

c.) A, G

d.) A, G, E

e.) None of the above

Complex Arrangement Example 2:

In which of the following colleges does Cstudy?

a.) X

b.) Y

c.) Z

d.) Either X or Z

e.) Cannot be determined

Complex Arrangement Example 3:

Who studies Chemical Engineering?

a.) A

b.) C

c.) B

d.) E

e.) None of the above

Complex Arrangement Example 4:

Which of the following combinations is correct?

a.) A-Civil-Z

b.) B-Chemical-Y

c.) C-Chemical-Z

d.) G-Electronics-Y

e.) None of these

In Consolidated Table Method, we prepare table like we did in Matrix method but the structure would be slightly different from the earlier one. Unlike, matrix method here, the second column is used to list down the objects that need to be arranged with the correct associated characteristics.

In this type, we don’t use any signs to denote the presence or absence instead the first column is used as eliminated option column. Using the information given above, here’ s the prototype of it.

Eliminated Options		Friends	Branch	College
Branch	College
		A
		B
		C
		D
		E
		F
		G

 

Now, like in the earlier method, we will also proceed similarly in this as well. We will first find out the directly associated features using the information given in the question and insert in the appropriate cell and put the other leftover eliminated options in the first column.

The direct Information given here is,

• D studies Electrical engineering in College 
• Fstudies Aeronautical Engineering in college Y with only 
• Estudies Computer Engineering and does not study in college X.
• Gstudies Electronics Engineering but not in college 

Using these points let’s fill in the inputs.

Eliminated Options		Friends	Branch	College
Branch	College
		A
		B		Y
		C
Mech, Chem., Civil, Comp., Aero., Electron	Y, Z	D	Electrical Engineering	X
Mech, Chem., Electrical, Electron., Civil, Aero	X	E	Computer Engineering
Mech, Chem., Electrical, Electron., Civil, Comp.	X, Z	F	Aeronautical Engineering	Y
Mech, Chem., Electrical, Civil, Comp., Aero.	X	G	Electronics Engineering

 

Now we can easily eliminate options of Electronics, Electrical, Computer and Aeronautical Engineering for A, B, and C.

Also,

• Adoes not study in college X and does not study Civil Engineering.

Hence our table becomes

Eliminated Options		Friends	Branch	College
Branch	College
Civil, Comp., Aero., Electron, Electrical	X	A
Comp., Aero., Electron, Electrical	X, Z	B		Y
Comp., Aero., Electron, Electrical		C
Mech, Chem., Civil, Comp., Aero., Electron	Y, Z	D	Electrical Engineering	X
Mech, Chem., Electrical, Electron., Civil, Aero	X	E	Computer Engineering
Mech, Chem., Electrical, Electron., Civil, Comp.	X, Z	F	Aeronautical Engineering	Y
Mech, Chem., Electrical, Civil, Comp., Aero.	X	G	Electronics Engineering

Since there should be at least 2 students studying in one college and according to the table formed using the information, D studies in X college and except C, there’s no student who is in X college and thus, we can indirectly infer that C is in X college. Also, none in college X studies Mechanical or Civil Engineering. Thus, C studies Chemical Engineering. Hence,

Eliminated Options		Friends	Branch	College
Branch	College
Civil, Comp., Aero., Electron, Electrical	X	A
Comp., Aero., Electron, Electrical	X, Z	B		Y
Comp., Aero., Civil, Mech., Electron, Electrical	Y, Z	C	Chemical Engineering	X
Mech, Chem., Civil, Comp., Aero., Electron	Y, Z	D	Electrical Engineering	X
Mech, Chem., Electrical, Electron., Civil, Aero	X	E	Computer Engineering
Mech, Chem., Electrical, Electron., Civil, Comp.	X, Z	F	Aeronautical Engineering	Y
Mech, Chem., Electrical, Civil, Comp., Aero.	X	G	Electronics Engineering

Since, A cannot study Civil engineering therefore, the only option left is Mechanical Engineering in college Z as it’s been clearly mentioned that F studies with only B in Y college. Therefore, we can also conclude that E and G also study in Z. Thus, the final table we will observe is

Eliminated Options		Friends	Branch	College
Branch	College
Civil, Comp., Aero., Electron, Electrical, Chem.	X, Y	A	Mechanical Engineering	Z
Mech, Chem., Comp., Aero., Electron, Electrical	X, Z	B	Civil Engineering	Y
Comp., Aero., Civil, Mech., Electron, Electrical	Y, Z	C	Chemical Engineering	X
Mech, Chem., Civil, Comp., Aero., Electron	Y, Z	D	Electrical Engineering	X
Mech, Chem., Electrical, Electron., Civil, Aero	X, Y	E	Computer Engineering	Z
Mech, Chem., Electrical, Electron., Civil, Comp.	X, Z	F	Aeronautical Engineering	Y
Mech, Chem., Electrical, Civil, Comp., Aero.	X, Y	G	Electronics Engineering	Z

Now, using the above table we can easily answer all the questions asked on the basis of information given. This is just one example there can be hundreds other and you can use either of the methods in the application according to your comfort. But, you will only get fluent with these kinds of problems if you practice them. As rightly said, “Knowledge is of no use unless you practice it’.

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All questions from CAT Exam Logical Reasoning and Data Interpretation
Logical Reasoning – Set 1: A high security research lab requires the researchers to set a pass key sequence based on the scan of the five fingers of their left hands.
Logical Reasoning – Set 2: There are 21 employees working in a division, out of whom 10 are special-skilled employees (SE) and the remaining are regular skilled employees (RE).
Logical Reasoning – Set 3: Eight friends: Ajit, Byomkesh, Gargi, Jayanta, Kikira, Manik, Prodosh and Tapesh are going to Delhi from Kolkata by a flight operated by Cheap Air.
Logical Reasoning – Set 4: tea taster was assigned to rate teas from six different locations – Munnar, Wayanad, Ooty, Darjeeling, Assam and Himachal.
Logical Reasoning – Set 5: Four cars need to travel from Akala (A) to Bakala (B). Two routes are available, one via Mamur (M) and the other via Nanur (N).
Data Interpretation – Set 1: Applicants for the doctoral programmes of Ambi Institute of Engineering (AIE) and Bambi Institute of Engineering (BIE) have to appear for a Common Entrance Test (CET).
Data Interpretation – Set 2: An old woman had the following assets: (a) Rs. 70 lakh in bank deposits
Data Interpretation – Set 3: Simple Happiness index (SHI) of a country is computed on the basis of three parameters: social support (S), freedom to life choices (F) and corruption perception (C).
Data Interpretation – Set 4: Funky Pizzaria was required to supply pizzas to three different parties.
Data Interpretation – Set 5: There were seven elective courses – El to E7 – running in a specific term in a college. Each of the 300 students enrolled had chosen just one elective from among these seven.

This post was contributed by Sejal Khurana. We are looking for contributions for our blog (paid). If you wish to write for us, please send an email to [email protected]

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