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How many ways can you distribute 4 different balls among 4 different boxes?

Apr 6, 2020

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The keywords noted here are different=distingusiable balls and boxes(you can tell the balls apart)...

Let's say the for balls have color white, yellow, red, and blue... and the boxes are labeled from box 1, box 2, box 3, and box 4.

Pick up the first white ball, I can place it in any of the four boxes, for 4 ways.

Next, pick up the yellow ball, I can place it in any of the four boxes again(since there is no restriction one ball per box)..

Similarly, the same flow for the red and blue balls..

Thus, the answer is 4*4*4*4=4^4=256 ways.

The choices are independent of each other...

Apr 7, 2020
edited by tertre  Apr 7, 2020

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How many ways can you distribute 4 different balls among 4 different boxes?

It sounds the same as putting 4 different people in a queue.

That is just 4! = 24

I have only considered the possibility of 1 ball per box.

If any number of the balls can be in any box the number is a lot bigger.

Tertre says it is 4^4. She may well be correct.    [4^4=256]

I counted with three balls and 3 boxes and her answer of 3^3 is definitely correct.

I have not worked through the logic of 4 balls though. I expect if is 4^4 just as Tertre claims.

one in each box = 4!= 24

2 in one and one in two others = 4C2*3*2*4=144

2 in one and 2 in another

= 2 in the red and 2 in one other = 4C2*3 = 6*3 =18

plus     none in the red, 2 in white, 2 in one other = 4C2*2  = 12

plus      none in red, none in white, 2 in blue and 2 in yellow = 4C2  = 6

18+12+6=36

3 in one and one in another = 4*4 *3= 48

4 in one box = 4

24+144+36+48+4 = 256

Tertre can you please spell out your logic to get this directly as  4^4

Apr 7, 2020
edited by Melody  Apr 7, 2020
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I am getting a different answer than @Melody.

I think it is 4^4=256 ways ( distinguishable balls and distinguishable boxes).

Apr 7, 2020
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Hi Tertre,

I would like you to think of a simplified problem.

3 balls numbered 1,2 and 3

3 boxes Red, white and blue

By my logic there will be 3!=6 possibilities

By yours, there will be 3^3 = 27 possibilities.

Now draw up a table with the headings, red, white and blue and see if you can work out which of ours is correct.

See if you can find more than 6 solutions   ;)

Melody  Apr 7, 2020
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Hello Melody,

Are there restrictions that there is one ball per box?

If there is a restriction, then yes, 3!=six(6) ways.

If there are no restrictions, ball one can go in the red, white, or blue box...this also happens for ball two and ball three...

Then, there are 3^3=27 ways.

Remember that the choices are independent.

tertre  Apr 7, 2020
edited by tertre  Apr 7, 2020
edited by tertre  Apr 7, 2020
edited by tertre  Apr 7, 2020
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Thanks Tertre,  you are right.

I had not thought of the possibility of having more than one ball per box.

Melody  Apr 7, 2020
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Tertre, I have worked through the logic and come to the same answer and you.

Assuming any number of balls can go into any box.

But although I also get 256  which is 4^4.

I do it the long way.

Can you please explain the logic behind going straight to 4^4.

Thanks :)

Melody  Apr 7, 2020
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The keywords noted here are different=distingusiable balls and boxes(you can tell the balls apart)...

Let's say the for balls have color white, yellow, red, and blue... and the boxes are labeled from box 1, box 2, box 3, and box 4.

Pick up the first white ball, I can place it in any of the four boxes, for 4 ways.

Next, pick up the yellow ball, I can place it in any of the four boxes again(since there is no restriction one ball per box)..

Similarly, the same flow for the red and blue balls..

Thus, the answer is 4*4*4*4=4^4=256 ways.

The choices are independent of each other...

tertre  Apr 7, 2020
edited by tertre  Apr 7, 2020
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Thanks Tertre, that was so obvious .. after you explained it

Melody  Apr 8, 2020
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Hi Melody:  I think he simply used the "Formula 1" from this site under "distinct Balls into distinct Boxes" :

https://www.careerbless.com/aptitude/qa/permutations_combinations_imp7.php

Apr 7, 2020
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Thanks very much guest.

I do almost all my probability from scratch but it is really handy to have such sites as a way of checking answers.

I have not seen that site before.

Tertre, is that what you did, or did you use your own logic?

Melody  Apr 7, 2020
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I think he simply used the "Formula 1" from this site under "distinct Balls into distinct Boxes" :

I think you are SIMPLY an ASSHOLE Mr. BB!!

Why yes, it’s very obvious that everyone accesses careerbless.com for combinatoric theory and formulas.

It couldn’t possibly be any of these:

http://www.elcamino.edu/faculty/gfry/210/DistributeDifBallsDifBoxes.pdf

https://math.berkeley.edu/~evans/Combinatorics

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter3.pdf

http://infolab.stanford.edu/~ullman/focs/ch04.pdf

...all of which are years older than the careerbless.com post.

It couldn’t possibly be the diligent study and practice in high school and the two or more years of university (probably in Tennessee); demonstrated by four years of practice in posting coherent solutions to questions on this forum.

Also, what makes you think Tertre is a “he”?

If you actually read her posts, you’d clearly see a feminine writing style.  This may be lost on you, Mr. BB. You have very narrow, unfocused observational skills.

Yep, you are SIMPLY an ASSHOLE Mr. BB!!

GA

GingerAle  Apr 7, 2020
edited by GingerAle  Apr 7, 2020
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trolololol!!!!! bb got trolololed!!!W! troll!!! troll mers gigner ale please troll!!! troll!!! me trolL!!!!! TROLOL!o!lolOlOLOLOL!!!

Guest Apr 7, 2020
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I'd really like to hear Tertre's thought processes here :)

You really do not need to be so aggressive Ginger.

Apr 7, 2020
edited by Melody  Apr 7, 2020
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deleted ..............

GingerAle  Apr 7, 2020
edited by GingerAle  Apr 7, 2020
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Actually, I do....

Historically, Mr. BB slights and trivializes the authors of competent posts: Alan and Heureka are two of the more notable targets.  He once managed to royally pissoff Alan. Not an easy task, considering that Alan is (probably) of Scottish heritage: Scotts are well known for their endurance and stoicism; just as we Irish are well known for our aggression and ass-kicking.

Mr. BB should be well known for being a blarney bag and an asshole.

GA

GingerAle  Apr 7, 2020
edited by GingerAle  Apr 7, 2020
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keep getting triggered snowflake

Guest Apr 7, 2020
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What triggered the melting of your snowflakes?

Was it your promotion from Blarney Bag to AssHole, or the fact that you can’t tell the difference between a boy and girl?

GingerAle  Apr 7, 2020
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how long did it take you to come up with that

Guest Apr 7, 2020
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About the same length of time it took for two snowflakes to melt.

GingerAle  Apr 7, 2020