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CAT 2021 Slot 1 — QA Question 8

Forming a committeeEasy

The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so that the groups always include 3 and 5, while 7 and 8 are never included together is

Answer & solution

Answer: 47

Solution

Easy

Numbers 33 and 55 are forced in. Among the four free numbers {1,2,4,6}\{1,2,4,6\} each is independently in or out. The restriction is only on 77 and 88 (never both), so split by how many of {7,8}\{7,8\} are chosen and keep the "three or more" size rule.

1

Fix the compulsory numbers. 33 and 55 are always present, so 22 members are already locked and the group size is automatically ≥2\ge 2. The free numbers are {1,2,4,6}\{1,2,4,6\} (each in/out, 24=162^4=16 choices) plus the constrained pair {7,8}\{7,8\}.

forced: 3,5free {1,2,4,6}:24=16 ways\begin{aligned} &\text{forced: }3,5\\ &\text{free }\{1,2,4,6\}: 2^4=16\ \text{ways} \end{aligned}
2

Case 1 — exactly one of 7,87,8 chosen. Choose which one (22 ways); the group already has 3,53,5 and one of {7,8}\{7,8\}, so size ≥3\ge 3 for every choice of the free four.

2×16=32 ways\begin{aligned} &2\times 16=32\ \text{ways} \end{aligned}
3

Case 2 — neither 77 nor 88. Group is 3,53,5 plus some subset of {1,2,4,6}\{1,2,4,6\}. To reach size ≥3\ge 3 we must pick at least one free number, so exclude the empty pick.

16−1=15 ways(drop the all-empty case, size 2)\begin{aligned} &16-1=15\ \text{ways}\quad\text{(drop the all-empty case, size 2)} \end{aligned}
4

Add the cases. (Choosing both 77 and 88 is forbidden, so no further case.)

32+15=47\begin{aligned} &32+15=47 \end{aligned}
Number of groups=47\text{Number of groups}=47
CAT 2021 Slot 1 QA Q8: The number of groups of three or more distinct numbers that can be chosen from 1, 2, 3, 4, 5, 6, 7 and 8 so th — Solution | TheCATExam