1. ## 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

thank you

2. ## 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?

3. ## Re: Combination or permutation question about sport balls

Originally Posted by bigmansouf
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. $^7\mathcal{P}_3\cdot^6\mathcal{P}_2\cdot ^5\mathcal{P}_1$

4. ## Re: Combination or permutation question about sport balls

18564 looks to be the answer.

5. ## 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.

6. ## 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.

7. ## Re: Combination or permutation question about sport balls

Originally Posted by romsek
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

8. ## Re: Combination or permutation question about sport balls

Originally Posted by Plato
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.

9. ## Re: Combination or permutation question about sport balls

Originally Posted by romsek
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

10. ## Re: Combination or permutation question about sport balls

Originally Posted by romsek
18564 looks to be the answer.

ignore ......

11. ## Re: Combination or permutation question about sport balls

Originally Posted by romsek
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