Questions tagged [combinations]
Combinations are subsets of a given size of a given finite set. All questions for this tag have to directly involve combinations; if instead the question is about binomial coefficients, use that tag.
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Permutations of n random variables using 3 alphabets [duplicate]
I am trying to work out the following:
The number of possible arrangements of (S1, S2, ... Sn), where each Si can take one of 3 values (A, B, C), and where adjacent Si's do not have the same value. ...
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A committee of 11 members is to be formed from 8 males and 5 females with at least 6 males in the committee [duplicate]
Q) A committee of 11 members is to be formed from 8 males and 5 females. The number of ways the committee is formed with at least 6 males is?
The general solution I see everywhere is to take 3 cases ...
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Probability of 2 individual people being put in the same group of 3 with 39 total people
I am trying to understand the possible number of combinations for two specific people (let's call them Person A and Person B) to be put in the same group of 3 if there are 39 total people to sort into ...
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I do not understand why do we divide with the k! that is used in nCk when multiplying two combinations or more combinations that has the same k value
I recently started studying Permutations and Combinations and following is one interesting question I came across.
There are 8 students in a class. The class ...
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Counting Permutations with Product Rule
I am working on my homework and I solved this problem and I just wanted someone to double check to see if my answer makes sense because I don't trust chat gpt.
5.4.3: Lining up club members for a ...
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a group of 20 people, 12 men and 8 women, must elect a board of 4 people. On how many ways can this happen if the board is to have at least 1 woman?
a group of 20 people, 12 men and 8 women, must elect a board of 4 people. On
how many ways can this happen if the board is to have at least 1 woman?
I know that the amount of combinations as a whole ...
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intuitively Understanding meaning of a Combinatorics problem to reach solution
I have been recently taking course on Combinatorics and landed on following problem, Here is the formal statement of problem:
A room contains a single bulb and $(2^{2^{10}}+2^{2^9})$ identical ...
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The interpretation of 'at least one' event in probability
In the following question, I don't understand why combination such as {20,19,18} is not even considered? Or it's already implied in some of the calculation below implicitly? (but it's not clear to me, ...
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Why is the count of unique results from rolling two dice once 7C2?
By "unique results" I mean that the order of the two resulting numbers does not matter - a 1 and a 2 is the same as a 2 and a 1. I have used a spreadsheet to brute force the answer of 21, ...
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Discrete math example clarification (stars and bars)
Example 6.3.4 from Discrete Mathematics by Richard Johnsonbaugh 8th ed, page 285 (abridged):
"Consider three books: a computer science book, a physics book, and a history book. Suppose that the ...
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How many matches can be made
Two schools should play one against the other table tennis. Five students represent each of
these schools. Doubles should only play. Each pair from one school should play against
each pair from the ...
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The Collector’s Problem
I have the following Question in my text book , where the Answer is given. I have got my own alternative Answer. I want to know whether that alternative Answer is Correct & Equivalent to the text ...
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Probability of exactly $m$ bins being occupied
N indistinguishable balls have to be assigned to K bins (where each bin can have at most C balls). What is the probability that exactly m bins are occupied?
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There are eighty items grouped into twenty. Each group contains 4 items. In how many ways can I combine 10 groups and the code to list them [closed]
In how many ways can 4items in each group be combined with 4 items of 9 other groups in the total of the eighty items
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The table top is in the shape of a regular heptagon and diagonals are drawn on it. In how many ways can the figure be colored with two colors?
The table top is in the shape of a regular heptagon and diagonals are drawn on it as shown below:
In how many geometrically different ways can the resulting figure be colored with two colors at our ...
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Ways to pick $k$ objects from $n$ objects of $p$ different kinds, where some are alike? [duplicate]
How many ways are there (in both case where order matters (permutations) and order does not matter (combinations) ) if one
pick $k$ candidates from $n$ elements of $p$ different kinds where:
$n_1$ are ...
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In how many ways can the passengers be conveyed to the neighboring village from the train station?
25 passengers arrive at a railway station and proceed to the neighboring village. At the station there are 2 coaches accommodating 4 each & 3 carts accommodating 3 each. Find the number of ways in ...
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Find the number of nonnegative integer solutions combination and deck of cards question [closed]
I have a question about combinations from my homework.
Question 1. Find the number of nonnegative integer solutions to $$(x_1+x_2+x_3)(x_4+x_5+x_6) = 20$$
How do I tackle this question? From my ...
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Distributing $m\cdot n$ identical items into $n$ distinct boxes, with $m$ distinct positions in box
There are $n$ boxes and $m\cdot n$ identical items. Each box has $m$ positions where an item can be placed. The order matters here. It is a difference if an item is placed in position 1, 2, or 3 or if ...
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Number of positive integral solution of $\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10$
I want to find the number of positive integer solutions of the equations given by
$$\sum_{i=1}^{10} x_i=30,\text{ where } 0 < x_i<7, \forall 1\le i\le 10.$$
I know the case that, for any pair of ...
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Unconventional Dice (PnC Question)
I am getting the answer as 48, but at some places, the answer given is 24. I am just fixing 1 at the top and 2 at the bottom. Rest 3,4,5,6 can be arranged in 3! ways and colours can be decided in ...
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In how many ways can four distinctive letters be posted in 6 post boxes such that any two go in same post box and remaining go to different boxes?
I have tried to solve it in two different ways. But I ended up with different answers.
First method
There are $\binom{4}{2}=6$ different $2$ letter combinations from a set of $4$ letters.
With the ...
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Probability that at least one bin has all balls of the same color
There are $32$ balls in total, equally split into red, green, blue, and white colors ($8$ balls each). I also have $4$ bins. What is the probability that in at least one of the bins, all the balls are ...
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Subsets of three numbers from a set of n numbers such that no two subsets have a shared pair of numbers
I've had this problem for a long time. The problem goes as follows (it is a bit hard to say concisely): Let's say, for example, I have the numbers 1-9 (n=9) and am looking to arrange these numbers in ...
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Probability: Card distribution problem
A deck of 52 cards is equally dealt to 4 players.
Find the number of ways to distribute the cards so that each player has exactly one card from each rank.
[Note: A deck of 52 cards consists of 13 ...
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Combination problems with replacement
A restaurant has n items on its menu. During a particular day, k customers will arrive and each one will choose one item. The manager wants to count how many different collections of customer choices ...
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Solving Combinatorial Problem - Red Ball Distribution in Three Bins
I'm reaching out to the community for help in tackling a combinatorial problem that I'm currently stuck on. The problem revolves around distributing twelve identical red balls into three bins, labeled ...
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What is the total number of rectangles in a grid that do not include a circle?
I am having trouble solving this question where I have to find the total number of rectangles in the grid shown below that do not include any of the circles.
I believe that this question can be done ...
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Find the number of non-empty subsets of $\{1,2,3,\dots,8\}$ which do not contain two consecutive numbers.
I obtained that for the set $A:=\{1,2\ldots,n\}$, the number of subsets $S$ of size $k$ such that no two elements of $S$ are consecutive numbers is $\binom{n-k+1}{k}$. Applying
$$\sum_{k=1}^{4} \binom{...
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Probabilities when two cards are drawn from a deck with $32$ cards
Would someone be kind enough to validate my reasoning when it comes to probabilities with a simultaneous draft of two cards in a 32 cards deck?
values : 1,7,8,9,10 J,Q,K
types : spade, heart, diamond, ...
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Probability with two different results
There are three universities: $U_1$, $U_2$, and $U_3$. $U1$ has three departments, $U_2$ has four departments, and $U_3$ has six departments. A tour is available that allows visitors to tour all the ...
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Expectation over vector coefficients defined in mixed deterministic-random protocol
Let $\textbf{x} = (x_1, \dots, x_n)$ a real vector,
We define the vector $\textbf{y} = (y_1, \dots, y_n)$ following this protocol:
We take a subset of $\{1, \dots, n\}$ of cardinal $n-l$, we denote ...
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In how many ways $n$ shoes can be arranged in a sequence so that the shoes of one pair do not stand next to each other?
In how many ways $n$ shoes can be arranged in a sequence so that the shoes of one pair do not stand next to each other?
I am sure that it can be done using inclusion-exclusion theorem. Therefore I ...
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Combinations from a set of disjoint sets?
Forall $1 \le i \le n$, let $A_i$ be a set of $r_i > 0$ distinct elements (s.t. all $A_i$ are pairwise disjoint.). Also let $m := \sum_i r_i$.
I am interested in choosing $k$ elements from $\...
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Using tricks to Maximize a Product of Binomial Coefficients $\binom{n}{x} \times \binom{n}{b - ax}$
I'm trying to find a solution to the following problem without resorting to brute force:
\begin{equation}
\text{maximize } \binom{n}{x} \times \binom{n}{y}
\end{equation}
subject to the constraints:
\...
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Number of colourings of the necklace.
I want to count the number of ways to color beads of a necklace green and red, such that two adjacent beads cannot both be red.
The necklace cannot be turned or reflected, the beads are labelled.
...
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Help me understand why my solution to this combinatorics problem is incorrect.
Question:
In how many ways a team of 11 players can be selected out of 15 players such that 2 particular players never come together in a team.
My solution:
No of ways = No of ways which p1 is ...
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Two congruent regular tetrahedrons are glued together by their bases. Count the ways to paint the faces of the resulting solid with $7$ colors.
Two congruent regular tetrahedrons are glued together by their bases. In how many geometrically different ways (we consider only rotations) the faces of the resulting solid can be painted (each with ...
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Probability that the first two and last two jellybeans agree in colors
A child has a jar of 6 red and 10 blue jellybeans. The child wants to
eat 4 jellybeans, so they grab 4 jellybeans one-by-one uniformly at
random without replacement from the jar. Find the probability ...
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Number of sequences $(A_1, A_2, . . . , A_k)$ of subsets of the set $[n]$ satisfying the condition $A_1 \cap A_2 \cap ... \cap A_k = \emptyset$
Determine the number of sequences $(A_1, A_2, . . . , A_k)$ of subsets of the set $[n]$ satisfying the condition $A_1 \cap A_2 \cap ... \cap A_k = \emptyset$
I think the answer would be:
We put each ...
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The formula expressing, the number of all sequences $(X_1, . . . , X_m)$ of length $m$ of subsets of the set $[n] = \{ 1, . . . , n \}$
Find the formula (may be in the form of a sum) expressing, for fixed natural numbers $m, n \geq 1$, the number of all sequences $(X_1, . . . , X_m)$ of length $m$ of subsets of the set $[n] = \{ 1, . ....
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Number of sequences $(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)$ with expressions from the set $Y = \{ 1, 2, 3, 4 \}$ that satisfy $2$ conditions.
Find the number of sequences $(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8)$ with expressions from the set $Y = \{ 1, 2, 3, 4 \}$ that satisfy (simultaneously) two conditions:
each number from the set $Y$ ...
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how long does it take to guess 4 digit pin when you have all 4 numbers
If I have a four digit pin, normally that would be $10000$ combinations ($10^4$), which would take a very long time to guess, however what if the person trying to access the pin had all four digits ...
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Difficulty in understanding Permutation
I have given a question of Permutation and Combinations along with its solution. I couldn't understand even a bit of it. I am an extreme beginner . A detailed explanation is highly appreciated
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Combinatorial proof $\sum_{k=0}^n {{n}\choose{k}}^2 \ m^k = \sum_{j=0}^n {{n}\choose{j}} \ {{2n - j}\choose{n}} \ (m-1)^j$
Combinatorial proof ($n > 0, m > 1$):
$$\sum_{k=0}^n {{n}\choose{k}}^2 \ m^k = \sum_{j=0}^n {{n}\choose{j}} \ {{2n - j}\choose{n}} \ (m-1)^j$$
I came up with a story that work only for one side ...
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Combinatorial proof: $\sum_{k=1}^n k^m = \sum_{i=1}^m {{n+1}\choose{i+1}} \ {m\brace i} \ i!$. I don't understand one part of it (I made it wrong).
Combinatorial proof (for: $n, m \geq 0$)
$$\sum_{k=1}^n k^m = \sum_{i=1}^m {{n+1}\choose{i+1}} \ {m\brace i} \ i!$$
The story: we have $m$ elements and $n$ empty buckets.
LHS:
We put every element of ...
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Calculate the combinations of 4 items in different number of sets [duplicate]
Let's say I have four colors. Red, Green, Blue, Black
I want to find all different combinations that those can be put to, being able to use sets of 4 colors, down to 1 color. Order will always be the ...
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Combinatorial proof verification: $\sum_{i=0}^n {{n}\choose{i}} { n-i\brack k} i! = { n+1 \brack k+1}$
I need to verify the following combinatorial proof:
$$\sum_{i=0}^n {{n}\choose{i}} { n-i\brack k} i! = { n+1 \brack k+1}$$
On the RHS we count all possible permutations of $n+1$ elements with $k+1$ ...
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Combinatorial proof $\sum^n_{k=0} {{2n}\choose{k}}{{n}\choose{k}}{{2n-k}\choose{n}} = {{2n}\choose{n}}^2$
I need to prove that:
$$\sum^n_{k=0} {{2n}\choose{k}}{{n}\choose{k}}{{2n-k}\choose{n}} = {{2n}\choose{n}}^2$$
So I figured that we can think of a situation in which we have $2$ groups of pairs of ...
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Repeated elimination of sets whitch contain a certain subset from a power set
I have the set S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
Basic combinatorics of no repetition and irrelavant order says the total number of subsets of S is the sum of binomial ...
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