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CAT 2018 Slot 2 · DILR · Networks and Routes · hard

CAT 2018 Slot 2 DILR: Social Network Friendships

Five friends — A, B, C, D and E — are part of a small social network in which "friendship" is always mutual (if X is friends with Y, then Y is friends with X). The table below shows how many friends each person has within this group of five.

| Person | Number of friends (within the group) |
|---|---|
| A | 3 |
| B | 2 |
| C | 3 |
| D | 2 |
| E | 2 |

Additional facts:
1. A and C are friends.
2. B and C are friends.
3. B and D are NOT friends.
4. A and E are NOT friends.

These facts are sufficient to determine every friendship in the group uniquely.

Question 1

Single correct option

How many friends do A and C have in common (i.e., how many people are friends with both A and C)?
  1. A. 0
  2. B. 1
  3. C. 2
  4. D. 3
Show answer and solution

Answer: (B) 1

Working out the full graph: A's friends must total 3 and already includes C; since A–E is not a friendship, A's third friend must be B or D. C's friends total 3 and includes A and B (given). Testing D as C's third friend forces contradictions with B and D's counts, so C's friends are A, B and E, and A's remaining friends are B and D (giving A: B,C,D). Full graph: A–B, A–C, A–D, B–C, C–E, D–E. Friends of A = {B,C,D}; friends of C = {A,B,E}. Common friend (excluding A,C themselves) = {B} only. Answer: 1.

Question 2

Single correct option

Which pair of people are BOTH friends with E?
  1. A. A and B
  2. B. B and D
  3. C. C and D
  4. D. A and D
Show answer and solution

Answer: (C) C and D

From the deduced graph (A–B, A–C, A–D, B–C, C–E, D–E), E's only friends are C and D. So the pair in which both members are friends with E is (C, D).

Question 3

Numerical answer

What is the total number of friendship pairs (edges) in this network of five people?
Show answer and solution

Answer: 6

The deduced friendships are A–B, A–C, A–D, B–C, C–E, D–E — 6 pairs in total. Check: sum of individual friend-counts = 3+2+3+2+2 = 12 = 2×6, confirming 6 edges.

Question 4

Numerical answer

How many of the five people have 3 or more friends?
Show answer and solution

Answer: 2

From the friend-count table, only A (3) and C (3) have 3 or more friends; B, D and E each have 2.
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Original question written in the CAT 2018 Slot 2 pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake