Grouping and Team Formations: Introduction

In reasoning based on Team Formation, we arrange the given set of objects or people in different groups or select objects/people to form a group or a team in accordance with the given constraints and conditions.

Here a few tips that can help solve these questions:

1. As you go through the clues, make their visual representation so that you don’t have to read again.

2. Keep track of the people/objects that should not be included or included together while forming a group or a team.

Let’s see some examples.

Grouping and Team Formations Question 1: Read the following information carefully and answer the questions based on it.

Two teams of five each must be selected from a group of ten persons – A through J – of which A, E, and G are doctors; D, H, and J are lawyers; B and I are engineers; C and F are managers. It is also known that

(i) Every team must contain persons of each of the four professions.

(ii) C and H cannot be selected together.

(iii) I cannot be selected in a team with two lawyers.

(iv) J cannot be in a team with two doctors.

(v) A and D cannot be selected together.

1. If C and G are in different teams, then who are the other team members of A?

A. C, D, E, and I

B. B, F, I, and J

C. B, C, H, and J

D. F, H, I, and G

2. Who among the following cannot be in the same team as I?

A. H

B. J

C. C

D. F

3. Who among the following must always be in the same team as A?

A. D

B. B

C. H

D. J

4. If F and G are in the same team, which among the following statements is true?

A. B and H would be in the other team (which does not include F and G).

B. E and I must be in the same team as F.

C. H must be in the same team as F and B must be in the other team (which does not include F).

D. C must be in the other team (which does not include F) and D must be in the same team as F.

Solution:

The team includes one person from each profession – Doctor, Lawyer, Engineer, and Manager. We have three doctors and three lawyers. Thus, one team would include two doctors and the other team would include two engineers.

Team Formations Introduction Example 1 Table (Part - 1)

> I cannot be selected in a team with two lawyers. J cannot be in a team with two doctors.

B and I must be in different teams are there are two engineers only.

Team Formations Introduction Example 1 Table (Part - 2)

> C and H cannot be together.

C is a manager and H is a lawyer. There are two possibilities.

C and F must be in different teams as there are two managers only.

Team Formations Introduction Example 1 Table (Part - 3)

> A and D cannot be selected together.

D is a lawyer. In case 1, D has to be in team 1, and in case 2, D has to be in team 2.

Team Formations Introduction Example 1 Table (Part - 4)

> In case 1, E and G must be in team 1, and in case 2, E and G can be in either of the teams.

Team Formations Introduction Example 1 Table (Part - 4)

Now let’s answer the questions:

1. If C and G are in different teams, then Case 2(b) would be true. The other team members of A would be F, H, I, and G. So, D is the correct option.

2. In all the cases, I and J are in the different teams. So, B is the correct option.

3. H must always be in the same team as A. So, C is the correct option.

4. If F and G are in the same team, then Case 2(b) would be true. H is in the same team as F and B is in the other team. So, C is the correct option.

Grouping and Team Formations Question 2: Read the following information and answer the questions.

A committee of four is to be selected from the nine eligible candidates.

The committee has the following positions General Secretary, Sports Head, Magazine Head, and Treasurer.

Conditions for selection of the committee of four are:

(i) Balwinder and Nishita should be selected together, but Balwinder can either be selected as Sports Head or Treasurer.

(ii) If Vyoma, Nishita or Durga are to be selected, then they should always be selected as Sports Head.

(iii) Swami and Durga should not be in the same committee.

(iv) Yuvan and Aiyaz should always be selected together; Aiyaz should never be selected as Treasurer.

(v) If Jia is selected, then Swami cannot be selected.

(vi) Aiyaz and Durga should be selected together.

(vii) If Aahan or Yuvan is selected, then they should be selected only as General Secretary.

1. Among the below-mentioned groups, if one person is removed from the group, then which group becomes a perfect committee according to the conditions stated above?

A. Nishita, Yuvan, Durga, Aahan, and Balwinder

B. Vyoma, Swami, Yuvan, Aiyaz, and Durga

C. Nishita, Balwinder, Yuvan, Aiyaz, and Swami

D. Nishita, Swami, Balwinder, Aahan, and Jia

2. Who among the following, if selected in the committee, will always be a magazine head?

A. Jia

B. Aiyaz

C. Balwinder

D. Durga

3. Who among the nine candidates cannot be selected in any committee?

A. Vyoma

B. Durga

C. Nishita

D. Balwinder

Solution:

> From Clue (ii), we can infer that only one of Vyoma, Nishita and Durga can be included in the committee. The fourth possibility would be none of them being included.

Team Formations Introduction Example 2 Table (Part - 1)

> Clue (i): Balwinder and Nishita should be selected together, but Balwinder can either be selected as Sports Head or Treasurer.

Since Nishita can only be Sports Head, Balwinder would be selected as Treasurer.

Team Formations Introduction Example 2 Table (Part - 3)> Clue (iii): Swami and Durga should not be in the same committee.

Team Formations Introduction Example 2 Table (Part - 2)

> Clue (iv) and (vi): Yuvan and Aiyaz should always be selected together; Aiyaz should never be selected as Treasurer. Aiyaz and Durga should be selected together.

This implies Yuvan, Aiyaz, and Durga must be selected together.

Team Formations Introduction Example 2 Table (Part - 4)

We have to leave out 5 candidates only. So, case 4 is not possible.

Team Formations Introduction Example 2 Table (Part - 5)

> Clue (v) and (vii): If Jia is selected, then Swami cannot be selected. If Aahan or Yuvan is selected, then they should be selected only as General Secretary.

The other three candidates in case 1 could be Swami, Jia, and Aahan. Since Swami and Jia cannot be together, this case is not possible.

In case 2, one of Swami and Jia would be included and the second member would be Aahan as General Secretary.

In case 3, Yuvan would be General Secretary. Since Aiyaz cannot be selected as Treasurer, he would be selected as Magazine Head. Jia would be selected as Treasurer.

Team Formations Introduction Example 2 Table (Part - 6)

Now let’s answer the questions:

1. Option (D) is correct. If we remove any one of Swami or Jia in option (D), the remaining group satisfies all the conditions.

2. Among the given options, Aiyaz, if selected, will always be Magazine head. So, B is the correct option.

3. Vyoma cannot be selected in any committee. So, A is the correct option.

Grouping and Team Formations Question 3: Answer the questions based on the following information.

K, L, M, N, P, Q, R, S, U, and W are the only ten members in a department. There is a proposal to form a team within the members of the department, subject to the following conditions:

(i) A team must include exactly one among us P, R, and S.

(ii) A team must include either M or Q, but not both.

(iii) If a team includes K, then it must also include L and vice-versa.

(iv) If a team includes one among S, U, and W, then it must also include the other two.

(v) L and N cannot be members of the same team.

(vi) L and U cannot be members of the same team.

1. What would be the size of the largest possible team?

A. 8

B. 7

C. 6

D. 5

E. Cannot be determined

2. What could be the size of a team that includes K?

A. 2 or 3

B. 2 or 4

C. 3 or 4

D. Only 2

E. Only 4

3. In how many ways a team can be constituted so that the team includes N?

A. 2

B. 3

C. 4

D. 5

E. 6

4. Who cannot be a member of a team of size 3?

A. L

B. M

C. N

D. P

E. Q

5. Who can be a member of a team of size 5?

A. K

B. L

C. M

D. P

E. R

Solution:

> Clue (i): A team must include exactly one among us P, R, and S.

Case 1: P

Case 2: R

Case 3: S

> Clue (ii): A team must include either M or Q, but not both.

Case 1: P, M/Q

Case 2: R, M/Q

Case 3: S, M/Q

> Clue (iv): If a team includes one among S, U, and W, then it must also include the other two.

This implies U and W cannot be included in Case 1 and Case 2. But both of them must be included in Case 3.

Case 1: P, M/Q

Case 2: R, M/Q

Case 3: S, M/Q, U, W

> Clue (iii), (v) and (vi): If a team includes K, then it must also include L and vice-versa. L and N cannot be members of the same team. L and U cannot be members of the same team.

L and U cannot be in the same team. This implies L won’t be included in case 3, but it may be included in case 1 and case 2.

Therefore, all the possible different combinations of teams are:

Case 1(a): P, M/Q

Case 1(b): P, M/Q, L, K

Case 1(c): P, M/Q, N

Case 2(a): R, M/Q

Case 2(a): R, M/Q, L, K

Case 2(c): R, M/Q, N

Case 3(a): S, M/Q, U, W

Case 3(b): S, M/Q, U, W, N

Now let’s answer the questions:

1. The size of the largest possible team is 5 which happens in Case 3(b). The members of this team would be either S, M, U, W, N or S, Q, U, W, N. So, D is the correct option.

2. The size of the team that includes K (Case 1(b) and Case 2(a)) can only be 4. So, E is the correct option.

3. There are 6 ways in which a team can be constituted so that it includes N. See the Cases 1(c), 2(c), and 3(b). Each case gives rise to two possibilities, one in which M is included, and the other in which Q is included. So, E is the correct option.

4. A team consists of a minimum of 2 members (one from P, R, and S and the other from M, Q). To include L, we must also include K, which increases the size of the team to 4. So, L cannot be a member of team of size 3. So, A is the correct option.

Similarly, S, U, and W also cannot be a member of a team of size 3. This is because if a team includes one among S, U, and W, then it must also include the other two.

5. Among the given options, only M can be a member of a team of size 5. So, C is the correct option.

Your Turn:

Let’s practice what we have learned. Read the following information and answer the questions.

Grouping and Team Formations Question 4: Two families are planning to go on a canoe trip together. The families consist of the following people: Robert and Mary Handerson and their three sons Tommy, Dan and William, Jerome and Ellen Penick and their two daughters Kate and Susan.

There will be three canoes with three people in each canoe. At least one of the four parents must be in each canoe. At least one person from each family must be in each canoe.

1. If the two mothers ride together in the same canoe and the three brothers each ride in a different canoe, which of the following must be true?

A. Each canoe has both males and females in it.

B. One of the canoes has only females in it.

C. One of the canoes has only males in it.

D. The sisters ride in the same canoe.

2. If Ellen and Susan are together in one of the canoes, which of the following could be a list of the people in another canoe?

A. Dan, Jerome, Kate

B. Dan, Jerome, William

C. Dan, Kate, Tommy

D. Jerome, Kate, Marry

3. If Jerome and Mary are together in one of the canoes, each of the following could be a list of people together in another canoe except

A. Dan, Ellen, Susan

B. Ellen, Robert, Tommy

C. Dan, Ellen, William

D. Dan, Tommy, William

4. If each of the Henderson children rides in a different canoe, which of the following must be true?

I. The Penick children do not ride together.

II. The Penick parents do not ride together.

III. The Henderson parents do not ride together.

A. Only I

B. Only II

C. I and II

D. I and III

1.

The arrangement would be as follows:

Canoe 1: M1, M2, B1

Canoe 2: F1, B2, D1

Canoe 3: F2, B3, D2

‘M’ donate mother, ‘F’ donate father, ‘B’ is used for the three brothers, and ‘D’ is used for Jerome and Ellen’s daughters.

Let’s analyze the options:

A. It is true that each canoe has both males and females in it.

B. No canoe has only females in it.

C. No canoe has only males in it.

D. The sisters ride in different canoes.

Therefore, A is the only correct option.

2. 

Remember that one person from each family must be in each canoe. Therefore, Jerome and Kate would be in separate canoes.

Canoe 1: Ellen, Susan

Canoe 2: Jerome

Canoe 3: Kate

Also, at least one of the four parents must be in each canoe. Therefore, one of Robert and Mary must be in Canoe 3.

Canoe 1: Ellen, Susan

Canoe 2: Jerome

Canoe 3: Kate, Robert/Mary

Now let’s analyze the options.

> A and D:

Jerome and Kate must be in separate canoes. So, these options are rejected.

> B: Dan, Jerome, William.

It includes one parent, which is Jerome, and members of both the families are also in this canoe. So, this combination is possible.

> C: Dan, Kate, Tommy

One of Robert and Mary must be present in the canoe with Kate. So, this combination is not possible.

Therefore, option (B) is correct.

3.

Canoe 1: Jerome, Mary

Canoe 2: Ellen,

Canoe 3: Robert, Kate/Susan

Since one person from each family must be in each canoe, one of Kate and Susan must be with Robert in canoe 3. Therefore, the combination in option (B) is not possible. So, B is the correct option.

4. 

Canoe 1: B1

Canoe 2: B2

Canoe 3: B3

Let’s analyze each statement.

I. The Penick children do not ride together.

This statement is true because at least one parent must be present in each canoe.

II. The Penick parents do not ride together.

This statement is false. It is possible for them to ride together. One such arrangement could be:

Canoe 1: B1, Ellen, Jerome

Canoe 2: B2, Robert, Kate

Canoe 3: B3, Mary, Susan

III. The Henderson parents do not ride together.

This statement is true. Because if they ride together, then one of the canoes will have members from one family only which should not happen.

Therefore, both I and III are true. So, D is the correct option.