My friend shared with me a story that after losing to his SO at Yahtzee, before they put the game away he just randomly predicted he would roll four 5’s and a 1. He then got that roll and freaked out. I wanted to calculate exactly how likely that was but my combinatorics knowledge isn’t […]

Is it possible for a multiset to have a “negative” number of one or more elements? If so, how are such multisets defined, and what terminology exists for them?

Prove that for integers $n \geq 0$ and $a \geq 1$, $${n + a – 1 \choose a – 1} = \sum_{k = 0}^{\left\lfloor n/2 \right\rfloor} {a \choose n-2k}{k+a-1 \choose a-1}.$$ I figured I’d post this question, which was on an assignment I did, since I thought the solution was so nice.

Let $N$ be a multiset of $n$ distinct objects having the same multiplicity $k$. For instance, $N=\{a,\,a,\,b,\,b\}$ where $n=2$ and $k=2$. I was looking for the problem of counting the number of all the permutations of all the non-empty subsets of $N$. For the N in the above example, there are 18 permutations of the […]

We have a normal deck of $52$ cards and we draw $26$. What’s the probability of drawing exactly $13$ black and $13$ red cards? Here’s what I have so far. Consider a simplified deck of $8$ (with $4$ $B$’s and $4$ $R$’s), we have 6 permutations of $BBRR,RRBB,RBRB,RBBR,BRBR,BRRB$, each with probability $p=\frac{4^23^2}{(8*7*6*5)}$, therefore the overall […]

I am trying to define a distance $F(X,Y)$ between two multisets $X$ and $Y$. For each pair of $x \in X , y \in Y$ there exists a distance function $f(x,y)$ which takes the range of $[0,1]$. An additional requirement of $F(X,Y)$ is that $F=0$ if one of $X,Y$ is a multiple of the other. […]

I have the following problem from book “Introduction to Probability”, p.32 A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, the superdeck has 52 · 10 = 520 cards, with 10 copies of each card. How many different 10-card hands can be […]

Stirling numbers of the second kind S(n, k) count the number of ways to partition a set of n elements into k nonempty subsets.What if there were duplicate elements in the set?That is,the set is a multiset?

Let $P$ and $Q$ be two finite multisets of the same cardinality $n$. Question: How many bijections are there from $P$ to $Q$? I will define a bijection between $P$ and $Q$ as a multiset $\Phi \subseteq P \times Q$ satisfying the properties: $\Phi=\{(p_1,q_1),(p_2,q_2),\ldots,(p_n,q_n)\}$, $P=\{p_1,p_2,\ldots,p_n\}$, and $Q=\{q_1,q_2,\ldots,q_n\}$. For example, suppose $P=\{1,1,1,3\}$ and $Q=\{1,1,2,2\}$. Then there […]

Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this theorem is usually phrased more clumsily, without multisets: Any integer greater than 1 can be written as a unique product (up to ordering of […]

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