I have 20 identical balls. I want to put these 20 balls to 4 different boxes. In how many ways I can do it? (If necessary we can keep one or more boxes empty)

Given n that represents the maximum value of all the numbers we can pick from such that 1 <= pick <= n Given k that represents how many numbers we must pick. I am interested in finding the number of combinations that will have a non-decreasing order. For example with n = 10, k = […]

I am completely lost on how to achieve this. I have no idea where to start, nor do I know what to use to find to prove this problem. Can someone help me with this?

Apologies in advance as I’m a programmer, not a mathematician. I am working on organizing pair programming in teams. So I have a set of individuals, and I want to work out all the possible unique pairing combinations so that we can have a set of pairs each day that ensures that no one re-pairs […]

When you have combinations where numbers are $0,1,2,\dots,m$, meaning we have $n=m+1$ and $k$, is there a way to see how $k$ of them sum up to a given number? For the sake of simplicity I have the numbers $0,1,2…,7$ (so $n=8$), and $k=3$. I need to find how much of these combinations with repetition […]

I’m trying to figure out a way to find the number of factors a number has based on its prime factorization without working out all the combinations. If the number breaks down into $n$ distinct prime factors I know it’s max number of combinations is $2^n$, with $\frac{n!}{(n-r)!r!}$ being the number of combinations when selecting […]

I’m working on this problem: An elevator in a building starts with 5 passengers and stops at seven 7 floors. If each passenger is equally likely to get an any floor and all the passengers leave independently of each other, what is the probability that no two passengers will get off at the same floor? […]

This was the question and answer I saw: How many different seven-card hands are there that contain two or more cards of the same rank? Solution: There are C(52,7) total hands. To subtract the ones that don’t have pairs, we observe that such hands have cards of 7 different ranks, and there are C(13,7) ways […]

We have a wall with $7$ slots. We can color the wall either with blue or red. How many combinations do we have to color the wall if two red slots cannot be neighbors? I thought, in a very intuitive way, to try all the different cases. For example if color $0$ slots of red, […]

So for example: How many permutations are there with 3 digits that add up to 4? For this question I just list all possible solutions: 004, 013, 022, 031, 040, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400 but what if the number of digit (k) is really big or the sum (n) […]

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