Discrete Math II - 6.5.3 Distributing Objects into Boxes

We complete section 6.5 by looking at the four different ways to distribute objects depending on whether the objects or boxes are indistinguishable or distinct. We finish up with a practice question. One thing to note is that while two of the instances do have a closed form expression, you should treat each question as individual problems and examine each for which strategies may work.

Video Chapters:
Intro 0:00
Recap of Combination and Permutation Formulas: 0:08
Distinguishable Objects and Distinguishable Boxes 0:50
Inistinguishable Objects and Distinguishable Boxes 4:02
Distinguishable Objects and Indistinguishable Boxes 5:27
Indistinguishable Objects and Inistinguishable Boxes 9:32
Your Practice 11:01
Up Next 14:04

This playlist uses Discrete Mathematics and Its Applications, Rosen 8e

Power Point slide decks to accompany the videos can be found here:

The entire playlist can be found here:

[Lobodog]

I've watched so many of your videos for my Discrete Mathematics course. Thank you for putting so much of your time and effort into making these videos. You have no idea how nice it is to know I can come to your channel and find what I need to have a good base for my learning.

[nguyentran]

For anyone who struggles to understand boxes or objects, you can try visualizing it with MS Paint or any painting tool you have. If a box/object is labeled/distinguishable you can pain each object with a different color, and the identical box/object you can put them on plain white. Then you can try putting each object into a box and see the outcome. Of course this only applies as learning experience and for small sets of boxes and objects.

For example for the practice b) when you put your colored balls into white boxes it doesn't matter which box you choose and the outcome will always be 5 boxes with their colored balls and 2 empty white boxes => 1 way.

[Monsta]

Thank you for helping me pass discrete math 1 and now discrete math 2 <3

[alexop]

For the distinguishable objects and indistinguishable boxes how about this example:
You have 4 additional ingredients for chocolate: almonds, peanuts, raisins, caramel. In the end you need to have a pack of 3 chocolates. All this additional ingredients must be used 0-3 times in chocolate. So question is how many types of sets of 3 chocolates can be.

[ernestodones]

I believe Practice a) problem is equivalent to finding all the 7-permutations of all the elements of the multiset M = {b1, b2, b3, b4, b5, 2*empty} instead of P(7, 5). Would you mind taking a look at it? (cuz i am not 100% sure if my reasoning is correct). Thanks for your videos btw, you are literally my combinatorics teacher.

[itamarhertz]

I fail to understand why the answer is 13 and not 12 in the question "How many ways are there to distribute 4 employees into 3 identical lockers, where each locker can be used for 0-3 employees." I tried using the formula x + y + z = 4 (3 lockers 4 people, 6c4 which is 15) and removing the options of 4 people in the same locker (3) getting 12.

[person]

I don't understand the last question. Why is the answer just 1 way.
There are 5 distinct balls & 7 identical boxes.
But I can still choose to put

Boxes: 1 2 3 4 5 6 7
Case1: 5 0 0 0 0 0 0
Case2: 4 1 0 0 0 0 0
Case3: 3 2 0 0 0 0 0
Case4: 2 2 0 0 0 0 0

So, aren't there 4 different ways?

Doesn't it matter if boxes have different no. of balls in them?
Or is it related to the line "Each ball must have At most 1 ball in it"? Does it mean we HAVE to fill 1 ball in each box if possible?