# Tagged Questions

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### How many Possible Combinations exist?

I have $120$ coins and $21$ buckets. Each bucket can hold $0$ to $20$ coins. How many possible coin/bucket combinations are there?
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### Simple König theorem

I have to prove the "simple" König theorem, without using the marriage theorem: Let $S$ be a set of size $mn$. Suppose that $S$ is partitioned into $m$ subsets, all having size $n$, in two ways: ...
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### If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same.

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same. In other words, putting a book in space 1 and a book in ...
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### Given a particular order how many times will it appear in all the possible permutations it has?

I have $10$ different coloured balls. I'm interested in selling them in packs of $15$ and the order is important. I know there are $10^{15}$ different ways of arranging these balls if I include the ...
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### Number of ways to divide variables into two categories

I'm looking for a possible solution to find out the maximum number of combinations that can be derived from the given variables. If I'm not mistaken, I think permutations and combinations is the way ...
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### distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
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### What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?

I worked out the following expression as the number of all possible "words" consisting of exactly $w$ letters from an alphabet $L$ of size $\left|L\right| = n \leq w$, and containing each of these $n$ ...
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### Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
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### Question of Permutation and combination

I have found a question from somewhere in the internet as follows: English language has 26 alphabets, out of 4 distinct vowels and 7 distinct consonants, how many letter patterns can be made ...
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### Equivalent of a sequence in regard to a certain length of a cycle for $\mathfrak{S}_{n}$

Let $n \in \Bbb{N}$ ( for me $0\notin \Bbb{N})$. Find the limit as $n$ tends to $+ \infty$ of the following sequence $$\frac{\alpha_{n}}{n}$$ where $\alpha_{n}$ is the number of permutations of ...
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### Number of license plates formed by four digits and one letter, qualified.

I need some help with this question: If a license plate for a vehicle consist of five characters: $4$ digits (the first of which cannot be $0$), followed by one letter of the alphabet (which ...