Question

There are cities A, B, C. Each city is connected with the other two by at least one direct road. A traveller can go from one city to another directly or via the third city. There are 33 total routes from A to B, and 23 from B to C. Find the number of roads from A to C.

• A. 6
• B. 3
• C. 5
• D. 10

Hint:

Translate route-count problems into equations using direct + via-third-city counts.

Answer 1

The Correct Option is C

Let roads $AB = x$, $BC = y$, $CA = z$. Given: total routes from A to B = direct $x$ + via C ($z \times y$) = 33: \[ x + zy = 33 \] Total routes from B to C = direct $y$ + via A ($x \times z$) = 23: \[ y + xz = 23 \] Also all $x,y,z$ are positive integers. Trial solving: subtract equations: $(x - y) + z(y - x) = 10 \ \Rightarrow\ (x-y)(1 - z) = 10$. From integer factorization and positivity, $x=3, y=8, z=5$ works. Thus $CA = z = 5$. \[ \boxed{5} \]