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

Guest Apr 6, 2020

#18**+1 **

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

#1**+2 **

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

**Edit: Added after Tertre's answer.**

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

256 is Tertre's answer too.

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

Melody Apr 7, 2020

#2**+1 **

I am getting a different answer than @Melody.

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

tertre Apr 7, 2020

#3**+1 **

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

#4**+1 **

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

#5**0 **

Thanks Tertre, you are right.

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

Melody
Apr 7, 2020

#6**0 **

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

#18**+1 **

Best Answer

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

#7**+1 **

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**

Guest Apr 7, 2020

#8**0 **

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

#9**-1 **

*“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

#11**0 **

I'd really like to hear Tertre's thought processes here :)

You really do not need to be so aggressive Ginger.

Melody Apr 7, 2020

#13**-1 **

**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