Team Formations (With Examples) | Logical Reasoning (LR) and Data Interpretation (DI) - CAT PDF Download

Introduction

In "Team Formations" questions, you're given a group of people or items along with specific conditions. Your task is to carefully understand these conditions and make choices to create the desired team. It's important to note down the key points of the conditions provided in the data. Sometimes, additional conditions may be introduced in the questions to be applied in certain situations. Therefore, it's essential to pay attention when reading these clauses and their relevance to the given scenarios.

Key Skills to Solve Questions

• The ability to visualize the structure of the teams to be formed including the number of people/objects in each team 
• The ability to order the clues in the correct order of usage (as explained in the theory of logical reasoning) 
• The ability to create symbolic representation of the various clues provided so that you can bring together each of the relevant clues while creating the teams 
• The ability to wait for and reach the appropriate time in the problem solving situation from where the indirect clues provided in the question can be used.

This topic generally deals with the selection of a team of say ‘r’ members from ‘n’ (n>r) available for selection or it can be the selection of committee of certain number of members.
Certain number of constraints drives this selection.

In order to understand these constrains and the implicit details related to them, let us start with an example.

Step Wise Solved Question 

Example: Among five students of group I – A, B, C, D, E and six students of group II – U, V, W, X, Y, Z, a team of five students is selected such that it consists exactly three students from group II. It is also known that:

• C and V cannot be selected together.
• If B is selected, neither U nor V can be selected.
• Among A,D,E and Y exactly two persons are to be selected.
• If E is in the team, at most one among U and W can be in the team.
• If A is selected, X has to be selected.
• Z will be in the team if and only if C is selected.

Find the total number of such teams possible.

Approach to solve

Let us take each condition one by one and try to get the obvious and hidden details about each information available.

The team consists of exactly three students from group II

We know that the number of students in the team is five. If exactly three students from group II is in the team then the rest two students must come from group I. This will always be the distribution in the team – two from group I and three from group II.

C and V cannot be selected together

• If C is in the team, V cannot be in the team.
• If V is in the team, C cannot be in the team.
• It does not mean that any one between C and V need to be selected. Both of them could simultaneously be out of the team

If B is selected, neither U nor V can be selected

• If B is in the team, U or V is not in the team.
• If U or V or both are in the team, then B is not in the team.
• All 3 can simultaneously be out of the team.

Among A, D, E and Y, exactly two persons are to be selected.

• We have to select any two among A, D, E and Y, it is mandatory.
• It can be – (A,D), (A,E), (A,Y), (D,E), (D,Y), (E,Y)

If E is in the team, at most one among U and W can be in the team.

• E and U can be in the team together.
• E and W can be in the team together.
• E can be in the team without U or W in the team.
• E, U and W cannot be in the team together.
• U and W can be in the team without E being in the team.
• All three can simultaneously be out of the team.

If A is selected, X has to be selected.

• If A is in the team, X has to be in the team.
• If X is in the team, A may or may not be in the team.
• If A is not in the team, X may or may not be in the team.
• If X is not in the team, A will not be in the team.

Z will be in the team if and only if C is selected

• This means if Z is in the team C is in the team
• If C is in the team, Z is also in the team.
• Both will simultaneously be in the team or out of the team.

‘If and Only If’ is the most important sentence out of as it implies that either select both or none, thus making them a pair.
Checking all the possibilities let us try to make all the possible teams.

Where to start

There can be a number of ways to start the puzzle. Three of the easiest approaches are:

Select two members from group I.

Group I has 5 members and we need to select 2 members from group I so total possible cases can be ⁵C₂ = 10. There is a possibility that not all of the 10 cases will result in a team and we have to solve individually for all 10 cases. For instance, we select B and C as two members from group I but from condition 3 we know that at least one among A, D or E (all three belongs to group 1) would be in the team thus making this case useless.

Select three members from group II

Selecting three from a group of 6 is ⁶C₃ = 20. Solving 20 cases would be even more time consuming.

Select two from condition 3.

We know that among A, D, E and Y, exactly two will be in the team. Number of possible cases are six as already discussed before. Solving 6 cases would definitely be less time consuming than other approaches. In addition, since we already abiding to one of the conditions so now we need to check for only five remaining conditions.

Let us start making teams by selecting two members as per condition 3.

Teams

• A and D are selected in the team (from condition 3), which means that the two members of group I are already selected and we need to select 3 members from group II. From condition 5, we can say that X will be in the team (A is selected then X is also selected). Y and Z cannot be selected (Y cannot be selected because among A, D, E and Y only 2 can be in the team and we already selected A and D; Z cannot be selected as C is not in the team (condition 6)). We are left with U, V and W. We need to select two among U, V and W so we can have three cases.
  
• Next we select A and E from A. D. E and Y. A and E belong to group I and so we need three more from group II. X will be one of them (condition 5). Now, since E is in the team either one of U or W can be in the team (condition 4). Y and Z cannot be in the team because of the reason explained in previous point. Therefore, we have two options. U and V or W and V.
  
• Now, we select A and Y. A belongs to group I and Y belongs to group II. We need one more member from group I and two more from group II. From group I, we can select B or C (D or E cannot be selected because of condition 3).
  With B, we cannot have either U or V in the team (condition 2) and since A is in the team, X has to be in the team. Without C, Z cannot be in the team. The only option left is W.
  With C, Z will also be in the team (condition 6). Thus, all 5 are selected and no more case possible.
  
• Next we select D and E. Both are from group I, leaving space for 3 group II members. Condition 4 allows only one among U or W because E is in the team. Again, Y and Z cannot be in the team (explained earlier). Thus, the only option left is with V and X.
  Note that X can be in the team even if A is not in the team.
  
• Next, we select D and Y. D is from group I and Y from group II. One more from group I can be B or C. Similar to the third case when we selected A and Y but here we are not restricted to the selection of X alone as there is no A. X will be in some of the teams but not in all teams. With C in the team, Z will also be in the team but V cannot be in the team (condition 1). With B in the team, neither U nor V can be in the team so we should select X and W.
  
• Last case is with E and Y selection. With E in the team, either U or W can be in the team. From group I we can select either B or C. Rest of the case is similar to the last case.
  

Solved Examples 

Example 1. 
Directions: Read the information given below and answer the questions.
A director is casting a movie about twins. Selection must be made from among nine people—Amartya, Bhupesh, Caruna, Divya, Elangovan, Farly, Girish, Harish, and Isha. Amartya is Bhupesh’s twin, Caruna is Divya’s twin, and Elangovan is Farly’s twin. Girish, Harish and Isha may each be selected only as “extras.”

• At least two pairs of twins must be selected.
• At least as many women as men must be selected.
• Amartya and Bhupesh must both be selected if either is selected.
• Caruna and Divya both must be selected if either is selected.
• Elangovan and Farly need not both be selected.
• At least one “extra” must be selected.
• Amartya, Bhupesh, Elangovan, Girish and Harish are men.

(i) Which of the following is an acceptable cast for the movie?
(a) Amartya, Bhupesh, Elangovan, Farly, Isha
(b) Caruna, Divya, Farly, Girish, Harish, Isha
(c) Bhupesh, Caruna, Divya, Elangovan, Harish, Isha
(d) Caruna, Divya, Elangovan, Farly, Girish, Harish, Isha

(ii) Which of the following people must be included in the cast?
(a) Divya
(b) Isha
(c) Farly
(d) Harish

To solve this we will create a diagram where men will be denoted by upper case letters and women with lower case letters and we will list the extras separately to avoid confusion. Here given that Caruna and Divya must both be selected because if they were not selected then Amartya and Bhupesh or Elangovan and Farly ( three of whom are men) would all be selected; as a result only two women Farly and Isha at most will be selected. However, according to the constraints at least as many women as men must be selected, thus Caruna and Divya must both be selected.

(i) Eliminate each options using the clues given in the information. Only option (d) suffices. Hence option (d) is the correct answer.
(ii) Caruna and Divya must both be selected. Hence option (a) is the correct answer.

Example 2. 
Read the information given below and answer the questions.
Vijayshree wants to take four courses this trimester. There are only seven courses in which she is interested: three marketing courses — Distribution, Advertising and PR; and four Finance courses— International Finance, Accounting, Corporate Finance and Financial Services. To meet college requirements she must take two marketing courses. There are some scheduling problems: International Finance overlaps both Advertising and Corporate Finance but she can choose Advertising and Corporate Finance as two different courses. Distribution is given at the same time as Accounting.
(i) If Vijayshree decides she will take International Finance, what will her other three courses be? 
(a) Distribution, PR and Advertising
(b) Distribution, PR and Financial Services
(c) Distribution, PR and Accounting
(d) PR, Advertising and Financial Services

(ii) If Vijayshree takes four courses this trimester, then which of the following statement is not possible?
I. She takes Accounting and does not take Advertising. 
II. She takes Corporate Finance and does not take Advertising. 
III. She takes International Finance and does not take PR.
(a) I only
(b) II only
(c) III only
(d) I and III only

(iii) Which of the following must always be true? 
I. Vijayshree must take PR if she takes Corporate Finance. 
II. Vijayshree must take Advertising if she takes Accounting. 
III. Vijayshree must take Accounting if she takes Advertising. 
(a) I, II, and III
(b) II, and III only
(c) I and II only
(d) II only

(i) It can be seen that if she takes International Finance, she would not be able to take the Advertising and Corporate Finance course. This means that she has to choose Distribution and PR as her two marketing courses and because Distribution overlaps with Accounting, her second finance course must be Financial Services. Option (b) is correct. 

(ii) Statement I is definitely not possible because if she takes Accounting she cannot take Distribution and she must have to take Advertising.
Statement II is possible, while statement III can be seen to not be possible as if she takes International Finance she has to take PR as Advertising would not be available to her.
Option (d) is correct. 

(iii) Statement I is not necessarily true as if she takes Corporate Finance she can take up Distribution and Advertising and skip PR, i.e., PR is not necessary if she has taken Corporate Finance.
Statement II is mandatorily true because if she takes Accounting, she cannot take Distribution and hence she would be forced to take up Advertising as one of her two compulsory marketing courses. Statement III is not necessarily true.
Thus option (d) is correct.