Question

How many ways are there to put 4 balls in 3 boxes if the balls are distinguishable but the boxes are not?

Answer 1

PLEASE HELP
WILL BE FAN AND GIVE MEDAL

Answer 2

for box 1, how many ways you can put a ball in?

Answer 3

How would you compute that?

Answer 4

you can put ball a, right?
you can put ball b, right?
you can put ball c, right?
you can put ball d also, right?
hence , there is 4 ways to put a ball ( any of them) into box 1, got it?

Answer 5

OK, got it

Answer 6

After then, you have only 2 boxes and 3 balls,
for box 2, exactly the same argument, how many ways you can put a ball into it?

Answer 7

3

Answer 8

yup, now the last one, how many ways?

Answer 9

2

Answer 10

yup, so, to get a ball inside a box, you have 4*3*2 =24 ways, right? not finish yet, just want you to get this part first

Answer 11

I'm afraid it's not that simple.

Answer 12

@Supreme_Kurt read my last comment :)

Answer 13

Carry on, L66, but you're off to a wrong start. Simply by saying "box 1" or "box 2", you've already made an erroneous step:

There is no box 1 or box 2 or whatever, the boxes are indistinguishable, remember?

Answer 14

And you can just call me Kurt :)

Answer 15

ok, I let you play this game @Supreme_Kurt
I want the Asker understands how I put the "indistinguishable balls" into indistinguishable box, then, permutation will be next.

Answer 16

So what is the right start Kurt?

Answer 17

Oh, sorry. My bad, then. I only realise that now

Answer 18

Kurt, compensate for "Oh, sorry. My bad...." continue the stuff, please, hehehe

Answer 19

OK, so what's next?

Answer 20

Let me just gather my thoughts.

Answer 21

Okay, looks like L66's approach was a bit dodgy after all.

Let's try reversing it.

Consider ball A, first. It can be put in any of the three boxes, right?
So how many possibilities is that?

Answer 22

3

Answer 23

That's right. Same goes for balls B, C, and D, right?

Answer 24

Yep

Answer 25

That makes 3^4 or 81 possibilities, that's IF the boxes are distinct from one another.

Answer 26

But they're not.

Answer 27

Catch me so far?

Answer 28

Yep!

Answer 29

Kurt??? Hello?

Answer 30

This is the tricky part. The part where we remove the distinction between the three boxes.

To understand that, let me draw you a picture. A sample way to distribute the balls between the boxes, while at the same time, giving labels to the boxes.