CAT 2025 Lesson : Selections - Case 6

Read the following and answer the questions that follow

Jonathan needs to form a team from 10 members. The team so formed needs to satisfy the following conditions.

1) Exactly one of A, B and C needs to be selected.
2) If B is selected, then D cannot be selected.
3) If one of C or E is selected, then the other should also be selected.
4) If F is selected, then G also has to be selected.
5) If H is selected, then A also has to be selected.
6) If one of I or J is selected, then the other cannot be selected.

15) In how many different ways can a 4-member team be selected with D as one of the selected members?

Answer:

16) What is the maximum possible number of selected members in the team?

Answer:

17) What is the maximum possible number of people in the team if B is selected?

Answer:

18) With I and J not selected, if the team consists of 5 members, then who among the following are definitely present in the team?

(1) F only
(2) D and F only
(3) D and G only
(4) D, F and G only

19) In how many ways can a 5 member team be formed with A as one of the members?

Answer:

Solution

15)
Notes:
(conditions given)
1. Exactly one from A/B/C & E
2. $B \varpropto D$
3. F & G
4. H & A
5. $I \varpropto J$

Case 1

For a four member team with D included, B cannot be selected. [Condition 2]
If A is selected, then C & E cannot be selected. [Condition 1]


H can either be selected or not selected which gives us three ways.

(i)
F & G cannot be selected but G alone can be selected which is accepted. I or J can also be included as the fourth member of the team instead.
A total of 3 ways possible -----(1)


(ii)
F & G can be selected which is 1 way to form the team-----(2)


(iii)
If F is not selected, then G can be selected. Among I and J, a team can be formed in 2 ways by including one of them.-----(3)


Case 2

For a four member team with D included,
B cannot be selected (Condition 2).
If A is not selected, then C & E is selected.
Since A is not selected, H is also not selected.
F & G cannot be selected as it would become a 5 member team.
One of I, J or G can be included in the team which gives us 3 ways.-----(4)


From 1, 2, 3 & 4,
the total number of ways in which a four member team can be formed is 9.

Answer: 9

16)
Notes:
(conditions given)
1. Exactly one from A/B/C & E
2. $B \varpropto D$
3. F & G
4. H & A
5. $I \varpropto J$

For a team of maximum members,

From condition 1, we should look out for pairs for maximum team members. Here C & E can be selected.
Since B is not selected, D is selected.
H cannot be selected as (H & A) have to be selected. [Condition 1 and 4]
F & G can be selected.
One among I and J can be selected.


Thus, a maximum of 6 members can be selected in a team.

Answer: 6

17)
Notes:
(Conditions given)
1. Exactly one from A/B/C & E
2. $B \varpropto D$
3. F & G
4. H & A
5. $I \varpropto J$

As mentioned, for a team with B, we cannot select A, C & E [Condition 1]
Based on condition 4, H cannot be selected.
For maximizing, F & G can be selected and one from I and J can be selected.


Thus a team of 4 can be formed with B included.

Answer: 4

18)
Notes:
(Conditions given)
1. Exactly one from A/B/C & E
2. $B \varpropto D$
3. F & G
4. H & A
5. $I \varpropto J$

As mentioned, a team of 5 to be formed with I & J not included

From condition 1 we can form 2 cases.

Case 1:

If A is selected then B, C & E are not selected.
Since B is not selected, D can be selected. $B \varpropto D$
Since A is selected, H is also selected.
F & G can be selected.


Case 2:
If A is selected then B, C & E are not selected.
Since B is not selected, D can be selected. ($B \varpropto D$)
Since A is not selected, H is also not selected. (H & A)
F & G can be selected.


Thus from both the cases D, F and G are common.
Based on the options, option 4 is the answer.

Answer: (4) D, F and G only

19)
Notes:
(Conditions given)
1. Exactly one from A/B/C & E
2. $B \varpropto D$
3. F & G
4. H & A
5. $I \varpropto J$

As mentioned, a team of 5 to be formed with A included B, C & E cannot be selected. Based on the selection of F, we can have two possibilities.

Case 1:

If F is not selected, then G can be selected.
D can be selected since B is not selected. ($B \varpropto D$)
H can be selected with no conflict.

One among I and J is to be selected, which gives us two ways of selecting a team of five.



Case 2:

If F is selected, then G should be selected. (F & G)
One among I, J, H or D should be selected.

Therefore, by combinations (selecting 2 out of 4 and subtracting the one way where I and J both are selected) ⁴C₂= 5 ways



From cases 1 & 2, we can conclude that there are a total of seven ways in which a team can be selected with A included.

Answer: 7

Answer:
15) 9
16) 6
17) 4
18) (4) D, F and G only
19) 7