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mathmath333

  • one year ago

Counting question

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  1. mathmath333
    • one year ago
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    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?

  2. ParthKohli
    • one year ago
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    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.

  3. ParthKohli
    • one year ago
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    Not six towns - I meant nine.

  4. mathmath333
    • one year ago
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    well i m yet to study probablity, is there any alternate way

  5. ParthKohli
    • one year ago
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    Also that should be \(4\cdot 3 \cdot \binom{3}2\)

  6. ParthKohli
    • one year ago
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    |dw:1434886089734:dw|

  7. ParthKohli
    • one year ago
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    We can divide the lines into two sorts: - Lines connecting towns within the same zone. - Lines connecting towns in different zones.

  8. mathmath333
    • one year ago
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    i counted lines in same zone , -> 36

  9. ParthKohli
    • one year ago
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    Now let's focus on any one zone. How many lines are there in one zone? There are three lines connecting every two towns, and there are exactly three couples of towns (towns A, B; B, C and A, C). So there are 9 lines within one zone. And so the total lines are 36.

  10. ParthKohli
    • one year ago
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    Now to calculate the lines that connect two towns not in the same zone, let's look at any town. There are exactly 9 towns connected to any town that do not belong to the same zone as the town, right? So there are 9 such lines for every town. Meaning that the total lines of this kind = number of towns * 9 = 12 * 9 = 108.

  11. mathmath333
    • one year ago
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    9*12=108 (why u took 12)

  12. ParthKohli
    • one year ago
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    Because there are 12 towns, haha.

  13. mathmath333
    • one year ago
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    ok ,answer given is 90

  14. ParthKohli
    • one year ago
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    Hmm, I don't see how. I'm sure we haven't double-counted anywhere.

  15. ikram002p
    • one year ago
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    |dw:1434887071634:dw|

  16. ikram002p
    • one year ago
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    like this ? |dw:1434887174189:dw|

  17. ikram002p
    • one year ago
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    |dw:1434887301011:dw|

  18. mathmate
    • one year ago
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    I would connect every town to each other by 1 line, which is the equivalent of a complete graph, giving 12*11/2=66 lines. Within each zone, to add the requirement of 3 lines between towns, we have to \(add\) two lines between towns, which is 3*2 lines per zone, or 24 lines for 4 zones. That gives a total of 66+24=90 lines.

  19. ikram002p
    • one year ago
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    lol so it would be 9*4+9*7=99 hmm

  20. ikram002p
    • one year ago
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    i must have 9 extra lines

  21. ikram002p
    • one year ago
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    ok i think i should have another way ;) can u imagine this as 3 D ?? |dw:1434887567296:dw|

  22. ikram002p
    • one year ago
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    this is how geometry fellow do it :P |dw:1434887663717:dw|

  23. ikram002p
    • one year ago
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    each point out side define a town each point inside define 3 towns each z define a zone

  24. ParthKohli
    • one year ago
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    Not sure why it's confusing... it was clear that we had to divide by two to avoid double-counting.

  25. ikram002p
    • one year ago
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    now first imply the triple points |dw:1434887838678:dw|

  26. ParthKohli
    • one year ago
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    That double-counting thing was intended as a hint. The reason why it is not 108 but 54 is that if A connects B, then B connects A.

  27. ParthKohli
    • one year ago
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    So the answer turns out to be 36 + 54 = 90

  28. ParthKohli
    • one year ago
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    This question reminds me of graph theory.

  29. ikram002p
    • one year ago
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    |dw:1434888136335:dw|

  30. ikram002p
    • one year ago
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    |dw:1434888298736:dw|

  31. ikram002p
    • one year ago
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    |dw:1434888512346:dw|

  32. ikram002p
    • one year ago
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    now my final answer :D |dw:1434888621084:dw|

  33. ikram002p
    • one year ago
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    dont thank me, it was really long time since i did graph theory :P

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