Counting question

Question

Counting question

mathmath333 · Answer

There are $12$ towns grouped into $4$-zones with $3$ towns per zone. It is intended to connect the towns with telephone lines such that every $2$ towns are connected with $3$ direct lines if they belong to the same zone, and with only $1$ direct line otherwise. How many direct telephone lines are required?

parthkohli · Answer

Nothing much to think about here. Consider one zone. If we look at the lines within that zone, there are $3\cdot \binom{3}{2}$ lines since there are 3 lines per two towns. 

Thus, overall, there are $3\cdot 3 \cdot \binom{3}{2}$ lines that connect towns in the same zone.

Now let's look at towns that don't belong to the same zone. For every town, there are six towns that do not belong to the same zone as that particular town, and thus, we find the number of towns and multiply that by 6 to find the lines that connect towns NOT belonging to the same zone.

parthkohli · Answer

Not six towns - I meant nine.

mathmath333 · Answer

well i m yet to study probablity, is there any alternate way

parthkohli · Answer

Also that should be $4\cdot 3 \cdot \binom{3}2$

parthkohli · Answer

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