COMBINATIONS with REPETITION - DISCRETE MATHEMATICS

We take a look at combinations with repetition, and discuss integer solution problems.

*--Playlists--*

*--Recommended Textbooks--*

Hello, welcome to TheTrevTutor. I'm here to help you learn your college courses in an easy, efficient manner. If you like what you see, feel free to subscribe and follow me for updates. If you have any questions, leave them below. I try to answer as many questions as possible. If something isn't quite clear or needs more explanation, I can easily make additional videos to satisfy your need for knowledge and understanding.

[JohnSmith-bfim]

Is it just me or is it like whenever you revisit permutation & combination it's like you're learning it for the first time

[SotosMoud]

I'm always amazed at the importance of 1. intuition and 2. breaking down a problem. No matter how complicated something may sound, it really is the simplest in its lowest level of complexity. The only thing differentiating is how many steps do you need to travel in order to reach this mind-blowing simplicity for everything to make sense, and start building from there. Thank you!

[joelbny]

Elegant explanation at 11:30 after the "shelves" model of how to solve that problem algebraically by substituting X3' = X3 - 2

[petergeorge]

wow. Great job! This was the most intuitive explanation i have seen for combinations w/repetition (replacement). I periodically find myself looking up permutations/combinations and i always forget the odd combination w/repetition formula; i say formula, because its this scenario that was never intuitive for me, so i just grab the formula. I have seen the 'divider' example and i guess it was always unnatural for me to consider dividers or non-bins, etc. Your example using 'replacement' will stick with me and is intuitive. Good job with shelves and books too! Thanks.

[KemiksPL]

I took me quite a while to get a grasp on the 4:00 problem. Here's my intuition:
You've got 7 items and 3 boxes. As long as you've got some items, you're choosing a box <=> you're choosing 7 times from 3 things with replacement <=> 3 + 7 - 1 choose 7.
I couldn't fully get that in the first explanation (around 5:00). I think that I was struggling with replacing the donuts intuition of "taking things" with "giving things away".

[adeebh.s]

The question at 3:02 asks to choose 12 DIFFERENT doughnuts from 20 there isn't it 20C1( the formula for without repetition)?
But everything else which you explained were pretty good, keep up the good work!

[SteveRayDarrell]

Great explanation. Most textbooks and tutorials I've seen dive right into technical proofs or worse just give the formula without addressing the intuition behind it.

[tonanbora]

+TheTrevTutor
Wait wait, @ 2:20
n+r-1 = 4+3-1 =*6* not 5
So it would be 6, chose 3 right?

[jessstuart]

I used a different method for the last problem.
You have 9+2 bins (9-stars, 2-bars) with the restriction (X3>1) that you cannot have bars in the 2 right-most bins.

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [no-bar ] [no-bar ]

You then have 9 bins where you can place 2 bars. C(9, 2). Just a different way of thinking about it.

[chasemarangu]

This helped me understand what a "combination with repeats allowed" was visually, and why the formula looked like that! Thanks for this!

5 letters A, B, C, D, E
₅P₃ number of 3 letter words not containing two of the same letter
5³ number of 3 letter words which can include e.g. AAA or CDC

₅C₃ number of sets of 3 separate letters
₅₊₃₋₁C₃ like the number of 3-letter words you can make, but AAC = ACA.

[vladnebotan]

In the last problem you can simply say that two balls have been reserved and simply calculate C(9, 2). In other words, You have to put 7 balls into three boxes but there are two balls in the third box already (the same thing you said in the video)

[lippy]

For the question at around 3:00 "In a donut shop, there are 20 types of donuts. How many ways can we select 12 different donuts to take home", I thought we should just be taking 20 choose 12? Since we are taking 12 different donuts, we wont be taking back the same ones...

[Microbiologist]

Thanks a lot! In my humble opinion the first explanation of combinations with repetitions is the most intuitively understandable. And I've seen lots of others explanations

[marcushendriksen]

At 2:27, shouldn't the coefficient be 6C3? After all, n+k-1 is equal to 6 when n is 4 and r is 3...

[Qizot]

Oh man, I've been loking for 2 days for a suitable explenation you you are the best so far. Great visualization, great thanks!

[Deksudo]

Alternative way to think about number of solutions to an equation (that's more similar to the methods used in previous videos of this playlist): You have 8 "slots" to place the separators, 6 of them in between the 7 ones and an aditional 2 at the beginning and the end. Like this: * _ * _ * _ * _ * _ * _ * _ * . You need to choose two of them and since you can choose the same slot twice, you use combinations with repetition. I.e. C(8+2-1, 2) = C(9, 2)

[benjamingarrard]

perfect explanation. This explains how n choose r works with repetition. Better concept than everything else I have seen.

[ruturaj_dm]

The example at Timestamp 3:45 is of Type 1) Combination, 2) No Repetition as 12 DIFFERENT DONUTS is mentioned. Can you explain why did you apply formula for Combination with Repetition?

[capybara]

shouldn't the second problem with x1+x2+x3=7 have an answer of 12 choose 10 since you're only accepting positive integer solutions, zero is not a positive integer

[kevinsantana]

great way of explaining it, knew how to use the formula but had no idea what the formula meant and how it was applied in finding total possible cominations. mank thanks!