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Thread: Combination or permutation question about sport balls

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    Combination or permutation question about sport balls

    Question;
    in how many ways can a boy arrange in a row 6 balls from 7 cricket balls, 6 tennis balls and 5 squash balls?

    i thought the answer would be $\displaystyle \frac{(18)(17)(16)(15)(14)(13)}{7! 6! 5!} $ but I was wrong
    I tried that fact that there are 3 unlike group of items that there are 3 ways of choosing which group to go first
    but from there i am a bit lost how to tackle this question
    Please can you help


    thank you
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    Re: Combination or permutation question about sport balls

    I believe what you have to do is determine every combination of $c, t, s$ (# of balls of each type)
    such that $c+t+s=6,~0\leq c \leq 6,~0\leq t \leq 6,~0 \leq s \leq 5$

    Having done that find $\dbinom{7}{c}\dbinom{6}{t}\dbinom{5}{s}$ for each combo and sum them all up.

    Are you expected to do this entirely by hand?
    Thanks from bigmansouf
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by bigmansouf View Post
    Question;
    in how many ways can a boy arrange in a row 6 balls from 7 cricket balls, 6 tennis balls and 5 squash balls?
    i thought the answer would be $\displaystyle \frac{(18)(17)(16)(15)(14)(13)}{7! 6! 5!} $ but I was wrong
    I tried that fact that there are 3 unlike group of items that there are 3 ways of choosing which group to go first
    but from there i am a bit lost how to tackle this question
    First question are the balls in the different sports considered identical?
    Say we have $CCCTTS$ that is three cricket balls, two tennis balls and one squash ball.
    In that case, that string can be arranged in $\dfrac{6!}{3!\cdot 2!\cdot 1!}$ ways.
    However look at this expansion: do you see the term $27x^6~?$ That tells us that there are twenty-seven ways to select six balls from the given collection of balls.
    BUT you will need to sit down and tabulate all twenty-seven compositions. That is a beast of a task.

    Now the only real difference if we consider all the balls as distinct,is to use permutations in stead of multi-selections.
    I.E. Say we have $CCCTTS$ that is three cricket balls, two tennis balls and one squash ball. $^7\mathcal{P}_3\cdot^6\mathcal{P}_2\cdot ^5\mathcal{P}_1$
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    Re: Combination or permutation question about sport balls

    18564 looks to be the answer.

    Combination or permutation question about sport balls-clipboard01.jpg
    Last edited by romsek; Jun 25th 2019 at 11:38 AM.
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    Re: Combination or permutation question about sport balls

    Oh bah... the problem calls for putting the balls in row, not in a bag.

    Ignore the previous posts.
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    Re: Combination or permutation question about sport balls

    Ok... as we have a row let's correspond this with a 6 digit base 3 number.

    Each digit specifies which type ball occupies that row position.

    The only glitch is that numbers can only have up to five of digit 2.

    So take all the possible, $3^6= 729$ possible arrangements and remove the ones with 6 2's.

    There's only one of them, all 2's.

    So there are 728 possible arrangements of the balls in a row.
    Last edited by romsek; Jun 25th 2019 at 02:13 PM.
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by romsek View Post
    Ok... as we have a row let's correspond this with a 6 digit base 3 number.
    Each digit specifies which ball occupies that row position.
    The only glitch is that numbers can only have up to five of digit 2.
    So take all the possible, $3^6= 729$ possible arrangements and remove the ones with 6 2's.
    There's only one of them, all 2's.
    So there are 728 possible arrangements of the balls in a row.
    You have two different models to consider:
    1) the balls in each sport are identical (all six tennis ball are indistinguishable).
    2) the balls are all unique.

    In many ways model #2 is the easiest to answer. If we have eighteen unique balls then
    $^{18}\mathcal{P}_6=13366080$. SEE HERE
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by Plato View Post
    You have two different models to consider:
    1) the balls in each sport are identical (all six tennis ball are indistinguishable).
    2) the balls are all unique.

    In many ways model #2 is the easiest to answer. If we have eighteen unique balls then
    $^{18}\mathcal{P}_6=13366080$. SEE HERE
    I'm assuming the balls of a given type are indistinguishable. Otherwise I think the problem would have just stated there were 18 balls.
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by romsek View Post
    I believe what you have to do is determine every combination of $c, t, s$ (# of balls of each type)
    such that $c+t+s=6,~0\leq c \leq 6,~0\leq t \leq 6,~0 \leq s \leq 5$

    Having done that find $\dbinom{7}{c}\dbinom{6}{t}\dbinom{5}{s}$ for each combo and sum them all up.

    Are you expected to do this entirely by hand?
    no we can use a calculator
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by romsek View Post
    18564 looks to be the answer.

    Click image for larger version. 

Name:	Clipboard01.jpg 
Views:	8 
Size:	106.1 KB 
ID:	39438
    ignore ......
    Last edited by bigmansouf; Jun 26th 2019 at 07:03 AM.
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    Re: Combination or permutation question about sport balls

    Quote Originally Posted by romsek View Post
    Ok... as we have a row let's correspond this with a 6 digit base 3 number.

    Each digit specifies which type ball occupies that row position.

    The only glitch is that numbers can only have up to five of digit 2.

    So take all the possible, $3^6= 729$ possible arrangements and remove the ones with 6 2's.

    There's only one of them, all 2's.

    So there are 728 possible arrangements of the balls in a row.
    thank you very much
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