# Tagged Questions

For questions about or related to multisets, a notion similar to sets with the difference that elements can be repeated.

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### Number of multisets with restrictions on specific element count

I am looking to find the number of multisets with restrictions on the number of specific elements. This isn't for homework, it is a work related problem. My set of items is {A, a, B, b}. I want to ...
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### Formula for combinations-

While I was thinking I found this formula: $\binom{n-k}{r-k} + \binom{n-k}{r-k +1} + \binom{n-k}{r-k +2} + ....+ \binom{n-k}{r-k +r}$ Where ...
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### Choose 3 letters.

Find in how many ways an arrangement of $3$ letters can be made from the $26$ different letters of the alphabet if any letter may be used once, twice or thrice. How many of these arrangements will not ...
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### Prove $\left(\dbinom nk \right)= \left(\dbinom{k+1}{n-1}\right)$ [closed]

I need to prove $\left(\!\dbinom nk \!\right)= \left(\!\dbinom{k+1}{n-1}\!\right)$ where the double parens denote multiset coefficients and $n,k$ are integers with $1 â¤ kâ¤ n$ using an algebraic proof. ...
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### Infinite Sets Proof [closed]

I have a few questions regarding this problem below: Prove that if A and B are finite sets, then A â B if and only if |A| = |B|. Would I assume that |A| = |B|? Which would obviously make A â B ...
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### Combination and Permutation S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}.

S= {A,B,B,C,C,C,D,D,D,D,E,E,E,E,E}. If I choose n element from S, how many possible combination (unordered) and permutation (ordered) are possible (without using decision tree or counting)? What is ...
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### Multiset Combination

How many Combinations can you make with the set {1,1,2,3,4} taken 2 at a time? If I do this in the way I do in Permutation: C(5,2) / 2!, I end up in wrong answer. Actually, there are 7 Combinations: ...
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### Given that the relation that aRb if and only if the smallest element of a is is equal to the smallest element in b?

X is the set of all nonempty subsets of the set {1,2,3,4,5,6,7,8,9,10}. a,b are elements of X. a) Find the number of elements in the equivalence class [{2,6,7}]? ...
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### Proving $\left(\binom{n}{k}\right)=\left(\binom{n-1}{0}\right)+\left(\binom{n-1}{1}\right)+\cdots+\left(\binom{n-1}{k}\right)$

Here, $\left(\binom{n}{k}\right)$ denotes the number of multisets in $N$ with length $k$. I can prove it using the fact that $\left(\binom{n}{k}\right) = \binom{n+k-1}{k}$ but I want another access. ...
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### Combinatorics Inclusion - Exclusion Principle

Find the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = 25$ with $1 \leq x_1 \leq 6, 2 \leq x_2 \leq 8, 0 \leq x_3 \leq 8, 5 \leq x_4 \leq 9.$ Firstly, I defined $y_i = x_i - lower bound$ ...
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### Number of ways to choose $4$ objects out of $6$ groups with $3$ members each

Suppose a box contains 18 balls number 1-6, three balls with each number. When 4 balls are drawn without replacement, how many outcome are possible? I took $6\choose4$ ways of picking the balls and ...
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### Enumerate possibilities when choosing exactly one from 5 of 6 subsets

The problem Given an arbitrary number n of sets of possibly different sizes, generate an m-column matrix where the rows describe all possible combinations of elements with one taken from each set. ...
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### How to do this simple set operation?

Suppose A and B are events with P(A) 0.4 , P(B) 0.6 and P(A and B) 0.25 . Calculate the probability P(A complement union B). A 0.25 B 0.65 C 0.75 D 0.85 What I tried?- P(A ...
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### Entropy of union of multisets

Assigning a random variable to some multiset: Assume that $S$ is a multiset. We can think of $S$ as independent sampling from some random variable. For instance, $S = \{H, H, T, T, T\}$ can be thought ...
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### Combinatorial Proof for Multiset Identity

$$\left(\!\!\binom{n}{k}\!\!\right)= \displaystyle\sum_{j=0}^k \binom{n}{j}\left(\!\!\binom{j}{k-j}\!\!\right)$$ Let $X$ be a set of $k$ element multi set of an n-element set. Let $P$ be a set of $j$...
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### How do I evaluate this combinatorically?

I recently came across this problem and couldn't even start on it. Would someone be able to help me? Given $m$ identical symbols, say H's, show that the number ...