CAT 1995 — QA Question 36
A, B, C and D are four towns, any three of which are non-collinear. Then the number of ways to construct three roads each joining a pair of towns so that the roads do not form a triangle is
Answer & solution
Correct answer: 24
- A
7
- B
8
- C
9
24
Let us choose a town, say A.
If I were to consider this as the base town and construct two roads such that I connect any pair of towns, I get
the following combinations:
1. AB – BC, 2. AB – BD, 3. AC – CB, 4. AC – CD,
5. AD – DB and 6. AD – DC.
From any of these combinations, if I were to construct a road such that it again comes back to A, then it would form a triangle.
To avoid a triangle, the third road that I construct should not be connected to A but to the third town.
Hence, the combination would be:
1. AB – BC – CD, 2. AB – BD – DC, 3. AC – CB – BD,
4. AC – CD – DB, 5. AD – DB – BC and 6. AD – DC – CB.
Thus, from each town, we can construct 6 such combinations.
Hence, total number of combinations that we can have from four towns = (6 × 4) = 24.