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Math Help - Counting Problems

  1. #1
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    Question Counting Problems

    A full set of tiles for the game Tantrix contains 56 hexagonal tiles. Each tile has three di erent coloured lines on it, from four possible colours: red, yellow, green and blue. There are 14 tiles with no blue lines on them, 14 with no red and so on.

    To commence a game, a player selects 6 tiles from the set of 56.

    i. In how many ways can the player select 6 tiles?
    ii. In how many ways can the player select 6 tiles with no blue on them?
    iii. In how many ways can the player select 6 tiles, each with a blue line?

    This is a game, please explain to me how to do it?
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  2. #2
    Junior Member utopiaNow's Avatar
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    Quote Originally Posted by melody View Post
    A full set of tiles for the game Tantrix contains 56 hexagonal tiles. Each tile has three di erent coloured lines on it, from four possible colours: red, yellow, green and blue. There are 14 tiles with no blue lines on them, 14 with no red and so on.

    To commence a game, a player selects 6 tiles from the set of 56.

    i. In how many ways can the player select 6 tiles?
    ii. In how many ways can the player select 6 tiles with no blue on them?
    iii. In how many ways can the player select 6 tiles, each with a blue line?

    This is a game, please explain to me how to do it?
    i. 56 ways to chose the first tile * 55 (since we already chose 1) ways to choose the 2nd. etc. to get:
    56*55*54*53*52*51 = answer, or 56P6 if you know nPr notation and meaning.

    ii. There are only 14 with no blue, therefore same method as i. except now we only have a total of 14 to choose from. Do you understand how to do this one?

    iii. Same method as i. and ii. except the total now is 56 - 14 = 42. 42 tiles have a blue line, since 14 out of 56 don't. So no our total is 42, so how many ways to choose 6 from 42? Follow the same method I outlined in i. except in i. the total we started with was 56.
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  3. #3
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    Quote Originally Posted by utopiaNow View Post
    i. 56 ways to chose the first tile * 55 (since we already chose 1) ways to choose the 2nd. etc. to get:
    56*55*54*53*52*51 = answer, or 56P6 if you know nPr notation and meaning.

    ii. There are only 14 with no blue, therefore same method as i. except now we only have a total of 14 to choose from. Do you understand how to do this one?

    iii. Same method as i. and ii. except the total now is 56 - 14 = 42. 42 tiles have a blue line, since 14 out of 56 don't. So no our total is 42, so how many ways to choose 6 from 42? Follow the same method I outlined in i. except in i. the total we started with was 56.
    ii. so 14*13*12*11*10*9 = 2162160 ways or 14P6 = same previous answer right?
    iii. so 42*41*40*39*38*37 = 3776965920 or 42P6 = same previous answer right?
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    Junior Member utopiaNow's Avatar
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    Quote Originally Posted by melody View Post
    ii. so 14*13*12*11*10*9 = 2162160 ways or 14P6 = same previous answer right?
    iii. so 42*41*40*39*38*37 = 3776965920 or 42P6 = same previous answer right?
    Correct. Make sure you understand why nPr or \frac{n!}{(n-r)!}, where n is the total number of elements available and r is the number of elements to be selected, gives you the correct answer.

    You might want to refer to: Permutation
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    Just a comment on the solutions posted thus far: They assume the order of the tiles matters, so, for example, the order in which the first 6 tiles is drawn is significant.

    If order doesn't matter, then you have a different problem and a different solution.
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