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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Product Rule proof confusion

I've got a textbook with this proof of the combinatorics product rule $$|S \times T| = |S| \cdot |T|$$ It says let $S = \{s_1,\ldots,s_m\}$ and $T = \{t_1,\ldots,t_n\}$. We induct on $n = |T|$. If $T =...
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Combinations and Permutations - Jury Service

Twenty people are called for jury service. Four of them are students, seven are waged employees, five are self-employed and four are retired. (a) How many different twelve-person juries can be ...
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1 vote
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How may ways can you rearrange the letters of MISSISSIPPI if the two P's must always stay together?

How may ways can you rearrange the letters of MISSISSIPPI if the two P's must always stay together? I considered ...
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0 answers
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Calculate the locations of an item in a symmetrical list

I am trying to calculate the position of a type of Item in a symmetrical or nearly symmetrical list. I have spent the last 20 minutes trying to describe the problem mathematically, but can't seem to ...
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0 answers
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is this a combination or permutation problem? arrange some items on a row

I found the following problem that says: If I have 5 cats and I want to take a picture of them in groups of three cats in a row, in how many ways can I put the cats for taking their picture? It is a ...
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1 vote
1 answer
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How to organise a robin robin where 8 teams play 8 games with each team playing each other team once plus one team twice with only 4 games at a time.

I am organising a sports day with 8 teams (T1-T8) and 8 activities (A1-A8) over 8 time slots (t1-t8). Each team can only do each activity once and can only play every other team once expect for ...
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-1 votes
1 answer
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Proof $\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}$ based on binomial coefficient definition

How can I prove this equation using the binomial coefficient definition? $\tbinom{n}{2}=\tbinom{k}{2}+k(n-k)+\tbinom{n-k}{2}, 1\le k\le n$ So far I've just written the equation using the ...
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3 votes
2 answers
87 views

How many ways can $7$ professors and $5$ students be seated at this long rectangular table so that no student sits across from another student?

So, I was doing a simple combinatorics problem, and I merely wanted to know where I actually went wrong. So the question was: My Method: So originally, I realised that the only possible combinations ...
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0 votes
1 answer
13 views

Calculate possible permutations/combinations of a set of unique values without repetition or reflection

Let's say I have a set of five unique values: {1, 2, 3, 4, 5} I want to calculate the number of possible combinations of these values without allowing any of the ...
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0 votes
1 answer
43 views

What's the number of combinations when randomly picking $n$ balls from a bag with infinite number of balls with 3 colors?

What's the number of combinations when randomly picking $n$ balls from a bag with infinite number of balls with 3 colors? Suppose 3 colors are A, B and C. I started with $n=1$ and the answer is 3 (A ...
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1 vote
1 answer
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Combinatorics Question - Group of 30 people.

Five people are randomly selected from 30 people, and then it is found that the height of these five people is increasing or decreasing. What is the probability that the height of the first person ...
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Counting number of ways to win a game

Tom and Jerry are in the finale of the game show. Currently, Jerry is leading and is a crowd favorite. The game show is such that all the points earned till the finale are considered and cumulated. ...
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-2 votes
0 answers
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Showing there are two student that have the same score [closed]

there are 15 students who take the exam for a course, the score scale is 0 to 100, the sum of all 15 students scores is 100. Show that there are two students who get the same score
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2 votes
1 answer
110 views

4 different colour balls, each of number four have to be arranged in a circular manner so that adjacent 3 balls are of different colour.

I have 4 red balls, 4 green balls, 4 Blue balls and 4 Yellow balls with me. I have to arrange them in a circular manner. The condition is that if we take any 3 adjacent balls, they should have to be ...
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1 vote
1 answer
82 views

Relation connecting $(3n)!$, $3^n$ and $n!$

Any idea on the relation for $(3n)!$ in terms of $3^n$ and $n!$ ? I have seen that there exist such relation for $(2n)!$ in terms of double factorials, ie. \begin{equation} (2n-1)!!= \frac{(2n)!}{2^n ...
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1 vote
2 answers
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Probability Combinations and Conditions

I have very limited math knowledge and might make little sense but here we go: If I wanted to substitute rolling a 4 sided die with "flipping" coins and adding a modifier, would it be unfair?...
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5 votes
1 answer
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How to analytically find the number of unique (simple) hydrocarbon chains, with arbitrary numbers of hydroxyl groups, of $N$ carbon atoms long

I wrote a Python script which already approximates this, however I would like to know how one can arrive at a purely analytical solution (or at least an analytically derived approximation). Let us ...
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0 votes
1 answer
35 views

Simplification of summation of combinations

Please refer below:- The equations are excerpted from IEEE magazine $ \mathbb {P}_{\mathrm {ra}} = \mathbb {P}\left ({ x > 0 }\right ) = 1 - e^{-\lambda }\; \tag{1}$ $\mathbb {P}_{0} = {\left ({ 1-\...
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1 vote
1 answer
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Birthday Problem: Finding a Probability Function of an Event

Problem: Ignoring leap days, the days of the year can be numbered $1$ to $365$. Assume that birthdays are equally likely to fall on any day of the year. Consider a group of $n$ people, of which you ...
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1 vote
0 answers
34 views

Algorithm to compute monomial coefficients from Vieta's Formulas

Let's say I know the $N$ roots $\boldsymbol r$ of a polynomial $p_N(x)$ and I want to compute the coefficients $\boldsymbol \alpha $ of the representation in monomials, i.e., $$p_N(x) = \sum_{j=0}^N \...
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-2 votes
0 answers
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In how many ways can a group of 5 people give a presentation?

I don't understand how 5 a group of 5 people can give a presentation. Tell me how many ways are there if the people have names and when they aren't named.
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-1 votes
2 answers
29 views

Probability of choosing r from n where order matters vs where order doesn't matter

In Ross's A First Course in Probability (8th Ed), there is an example where 5 people are to be selected from 6 men and 9 women. We want to know the probability of selecting 3 men and 2 women. The ...
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1 vote
2 answers
53 views

Three-of-a-kind Poker Hand Problem

Problem: Three-of-a-kind poker hand: Three cards have one rank and the remaining two cards have two other ranks. e.g. {2♥, 2♠, 2♣, 5♣, K♦} Calculate the probability of drawing this kind of poker ...
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0 votes
0 answers
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Unique combination when given a sum with fixed variable [duplicate]

Recently I learnt something about stars and bars knowledge.For example, if we want to find combination of 𝑥1+𝑥2=3, we can use 5C2, or 5!/(3!2!) = 10. Possible solutions includes : 0+0+3 0+3+0 3+0+0 ...
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5 votes
2 answers
98 views

Simplifying a sum of binomial coefficients multiplied by power of choice number

Let $n$ be a positive integer. Prove that $$ \sum_{k=0}^n \binom{n}{k}(k+1)^{k-1}(n+1-k)^{n-k} = (n+2)^n$$ Trying to generalize a combinatorial problem, my collegue have obtained the LHS with some ...
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0 votes
0 answers
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Drawing colored balls without replacement; tracking maximum balls picked across different colors

The following problem is a part of a research problem in a different field that I am working on. I have removed the details from the field on purpose because it is far removed from this piece as such ...
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-3 votes
1 answer
34 views

How many words can we make with the following letters $PTEXYPADFYOLNQYIG$? [closed]

How do you solve this? How many words can we make with the following letters PTEXYPADFYOLNQYIG keeping the vowels in the same position?
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  • 3
0 votes
0 answers
22 views

Combinations without duplicates of specific set elements

So, I have got a set with the following elements: {A1, A2, A3, B, C, D, E1, E2, F, G, H} and I have been asked to calculate the number of combinations with the subsets of 3, 4 and 5 elements where A1, ...
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4 votes
2 answers
1k views

A man desires to throw a party for some of his friends. In how many ways can he select 8 friends from a group of 14 friends?

A man desires to throw a party for some of his friends. In how many ways can he select $8$ friends from a group of $14$ friends if the two of his friends(say ’A’ and ’B’) will not attend the party ...
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  • 351
0 votes
0 answers
22 views

Finding the number of combinations for any total number T given by a sum of n five sided dices with side values ranging from 2 to 6.

The setting I know how to calculate the number of combinations for any given sum and side of a dice but i wanted to caculate the number of combination for any given sum with a dice starting not a one ...
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0 votes
2 answers
64 views

Algorithm to derive possible combinations of a set e.g., $A = [1, 2, 3, 4]$ and $k = 3$ and $L = [[1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4]]$

Given a set of numbers A and an integer k, I want to derive a list of sets L such that all the sets in L are the distinct combinations of the elements in A picking k at a time. For example: $A = \{1, ...
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-1 votes
0 answers
35 views

In how many ways can the committee be selected if the committee must have at least one member of each sex? [duplicate]

A committee of $5$ is to be formed from $5$ men and $5$ women. In how many ways can the committee be selected if the committee must have at least one member of each sex?
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2 votes
3 answers
80 views

Find $X/1430$ when $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$

Let $X=(^{10}C_1)^2+2(^{10}C_2)^2+3(^{10}C_3)^2+ ...+10(^{10}C_{10})^2$, then what's the value of $X\over1430$? I don't even know where to begin on this question. All solutions I've seen on various ...
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3 votes
1 answer
47 views

Is the following combinatorial relation correct?

I am confused regarding the following problem in combinatorics ( statistical mechanics ). Suppose I have the following relation : $$\sum_{i=1}^N n_i=\bar{N}$$ I have to find out the number of possible ...
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0 votes
1 answer
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Calculate Cov(X,Y) where X is #children with no books and Y is #children with exactly 1 book while distributing r books to n children.

Suppose $r\geq 1$ distinct books are distributed at random among $n\geq 3 $ children. Let $X$ be the number of children who do not get any book, and $Y$ be the number of children who get exactly one ...
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3 votes
0 answers
76 views

A tight upper bound on this Binomial sum

I have the following function: $P(n)=q^n\sum\limits_{H=0}^{n-1}{{H+n-1\choose H}w^H}+w^n\sum\limits_{H=n}^{\infty}{{H+n-1\choose H}q^H}$, where $0<q<0.5<w<1$ and $q+w=1$. My end goal is to ...
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0 votes
1 answer
60 views

Probability regarding yellow and white cabs and two independent witnesses

20% of the cabs are white and the other 80% are yellow. A cab was involved in an accident and ran away. An eyewitness to the accident claims that the cab was yellow. Knowing that eyewitness tell the ...
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0 votes
2 answers
128 views

In how many ways can a test paper of six questions be attempted, each question being of a TRUE/FALSE type?

In how many ways can a test paper of six questions be attempted, each question being of a TRUE/FALSE type? $0$ questions attempted out of $6$ then $0*0!=0$ $1$ Question attempted out of $6$ then $1*2!...
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  • 351
3 votes
0 answers
49 views

How to interpret these combinatorics equations' combination meanings

Let$$S_n:=\sum_{i=0}^n\frac{(-1)^i}{2i+1}\binom ni\\ H_n:=\sum_{i=0}^n\frac{(-1)^i}{2i+3}\binom ni$$then$$\begin{align}S_{n+1}-S_n&=\sum_{i=1}^n\frac{(-1)^i}{2i+1}\binom{n}{i-1}+\frac{(-1)^{n+1}}{...
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2 votes
2 answers
72 views

Ways to choose $6$ numbers from $\{1, \dots, 20\}$ so that at least four are odd

How many ways are there to choose $6$ numbers from the set $\{1, \dots, 20\}$ so that at least four of them are odd? I came up with two different answers to this question, and I can't see why either ...
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0 votes
1 answer
39 views

How do you calculate the possibilities of an alphanumeric string?

I want to know how many combinations are possible with a 4 character 4 digit string. All capital. Ex. ABCD1234, AAAA0000 - ZZZZ9999. What's the answer but more importantly, what's the formula? ...
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-1 votes
1 answer
40 views

Is there a way to represent a combination in an equation? [closed]

Is there a way to represent a combination in an equation? I am not familiar with this problem so bear with me. Say we have the following sequence: $1, 2, 3, \ldots, 61, 62, 63, 64$, hence there are $P=...
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0 votes
1 answer
62 views

Distribution of 20 fruits equally among 5 persons with some restrictions.

Find the number of ways in which $5$ Apples,$5$ Bananas,$5$ mangoes and $5$ Oranges (fruits of the same species are alike) can be distributed equally among five persons so that exactly $2$ of them get ...
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3 votes
2 answers
87 views

Probability of getting at least one pair in a combination

Assume that I have 6 symbols which I label $\{x_1,x_2,x_3,y_1,y_2,y_3\}$. Now I make a random 6-symbol combination with repetition, such that every combination has the same probability of occuring. ...
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-1 votes
0 answers
37 views

How many four-digit numbers can me formed using the digits $1,2,3,4,5$ when at least one digit is repeated? at least two digits are repeated?

How many four-digit numbers can me formed using the digits $1,2,3,4,5$ when at least one digit is repeated? at least two digits are repeated?
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0 votes
0 answers
20 views

combinatorial proof $ \binom{n+1}{r+1}=\binom{n}{r}+ \dots +\binom{r+1}{r}+ \binom{r}{r} $ [duplicate]

I need to proof the identity $$ \binom{n+1}{r+1}=\binom{n}{r}+ \dots +\binom{r+1}{r}+ \binom{r}{r} $$ I was able to give an "algebric" proof, but I was asked to give an combinatorial ...
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  • 1
1 vote
1 answer
32 views

Chance of getting a specific value from a combination with repetition

Let's say I have a set of $m$ distinct elements and I create a new set of size $n$ ($n\geq m$) choosing those elements by a combination with repetition. Then, I randomly draw a value from this new set ...
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  • 11
2 votes
3 answers
111 views

Placing different colors of indistinguishable balls around a circle

$3n$ indistinguishable balls are coloured with $n$ colours so that each colour is to be used exactly three times. In how many ways these coloured balls can be placed around a circle so that $3$ balls ...
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0 votes
0 answers
33 views

Discrete Mathematics Counting and Combinations/Variations Climbing Stairs

Let's say we have a staircase that has $10$ steps. As we walk up the staircase we must always end on step $10$. How many paths can I take up the staircase? Each path is distinguished by which steps I ...
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0 votes
1 answer
73 views

Discrete Mathematics Counting and Combinations/Variations on Chess Board

Knight Movement on Board for Questions Assume the white Knight starts on square b1 as shown in figure (a). How many paths can the Knight take to reach the b8 square marked ”X”? Assume that the knight ...
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