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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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1answer
24 views

Finding the number of ways to purchase the fruits.

There are three kinds of fruits in the market . How many ways are there to purchase 25 fruits from among them if each kind has atleast 25 of its fruit available?
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0answers
46 views

Finding the best possible combination of 4 variables to produce a successful outcome

I'm trying to find the best possible combination of 4 variable values to produce a successful outcome, with the data of only ~100 different recipe combinations. However, the correct outcome is a also ...
8
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3answers
310 views

What is the domain of a division of functions?

This question is about real functions of real variables. I think that, in general, if the domain of some function $f(x)$ is A, and the domain of another function $g(x)$ is B, then the domain of $(f/g)...
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1answer
44 views

Selecting unique items from combinations

Given a list of combinations (not permutations), how many do I need to randomly select to ensure $m$ unique items? For example, given combinations of $3$ letters, if I want $6$ unique items then it ...
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0answers
17 views

The number of ways to distribute N distinct balls on M distinct bins where M is greater than N [on hold]

Assuming that I have N Balls and M bins: Assume that N = 5 and M = 10 The number of balls is always less than or equal the number of bins. You can't put two balls in the same bin. The bins are ...
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1answer
22 views

combinations and permutations, how can I formulate it?

What mathematical formulation that can help me to generate all possible (N x N) matrices where the sum of all elements inside each matrix equal constant number M ? Note that: M > N If it can be ...
4
votes
1answer
57 views

Number of ways to roll $S$ with $n$ dice.

I have been working on this problem for the past 3 days and I am not having a lot of luck with it. I posted part of a) here already and I got some very useful advice. I am sitting a test in this ...
2
votes
1answer
36 views

how many words of $4$ consonants and $3$ vowels can be made from $12$ consonants and $4$ vowels? if all the letters are different?

my effort for this question is I am selecting $4$ consonants from $12$ available consonants and $3$ from $4$ available vowels. After selecting $4$ consonants and $3$ vowels now I have $7$ letters in ...
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2answers
32 views

Asymptotic behavior of combinations: approximating Hypergeometric by Binomial

A form of the hypergeometric distribution is $$P(X=x)=\frac{\binom{Np}{x}\binom{Nq}{n-x}}{\binom{N}{n}}$$ where $N\equiv$ total number of elements of the sample space $p\equiv$ probability of ...
2
votes
2answers
26 views

Sum of coefficients of high degree terms in multivariate polynomial expansion

I want to expand the following multivariate polinomial $$\left(\sum_{i=1}^{m} x_i\right)^{n}$$ where $m\geq n$ are both integers. For a fixed integer $k\in\{1,...,m\}$, how to find the sum of ...
1
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1answer
54 views

Number of Paths to get from One point to Another

What are the total number of paths that can be taken to get from point A to point B ? Rules :- 1)You can move up ,down ,left , right 2)You CANNOT return to a point that you have been to before ie no ...
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0answers
46 views

Combinatorics - How many numbers with all different digits are between $1000$ and $7856$?

To answer this question I was trying to solve the problem by parts. So, first I counted from 1000 to 7000: $6*9*8*7$ that means there are $6$ possibilities on position one, $9$ to position 2, $8$ to ...
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4answers
52 views

In how many ways can $4$ same oranges and $6$ different apples be distributed to $5$ distinct boxes?

If we have $4$ same oranges and $6$ different apples. In how many ways could we distribute them in $5$ different boxes? I have thought the first part of this problem as saying that the $4$ same ...
0
votes
1answer
47 views

Let S be a Set and let A be a subset of S. how many options there are to choose 2 subsets from S that their intersection is exactly A?

I'm struggling with this combinatoric problem - i marked the size of Set S as n and the size of A as k. I first thought of it this way: in order that the interaction of 2 subsets of S will be exactly ...
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1answer
24 views

Difference between permutation and combination formulas for repetition and not

In the theory of permutations and combinations there are several formulas which include permutations with repetition and without , same for combinations. I know the difference between permutations and ...
0
votes
1answer
27 views

Probability of winning with multiple tickets in multiple draws

Unfortunately, after hours of searching I was unable to find an answer to the following scenario: Let's assume I play five unique combinations in a weekly 6/49 lottery, where the chance of matching ...
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2answers
29 views

Number of ways of selecting a mathematics team and an informatics team with conditions

If we have $45$ different students ($20$ students from 12th grade, $25$ students from 11th grade). If we want to choose $15$ students for a mathematics team and $10$ students for am informatics team. ...
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0answers
14 views

Possible subset by selecting elements from the sets using GAP [closed]

I have a group of order n and k conjugacy classes of different order. I want to construct a set of a specific cardinality by selecting elements from the conjugacy classes. In how many ways a set of ...
1
vote
1answer
77 views

How many sequences length $n$, taken from $\{1,2,3,…,k\}$ that the sum of the $n$ elements in the sequence will be divisible by $k$.

I wonder if you can help me with this question I am being dealing with. My line of thinking was this: I know that the sequence is of length $n$, so I divided it into $n$ cells. $$x_1+x_2+x_3+\dotsb ...
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0answers
48 views

Sum of combination

I need to know if we can derive any formula to calculate this $$\sum_{i=k}^n\binom{n}{i}$$ I don't know if this question has been asked or not (or I didn't search for the correct keyword). While ...
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2answers
36 views

Number of ways to find $4$ positive integers less than or equal to $25$ with gap of at least $3$ between those integers

In how many ways you can make an unordered selection of four positive integers each less than or equal $25$ such that every pair of selected integers differ by at least $3$, i.e. for any two such ...
3
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6answers
874 views

You flip a coin $10$ times. How many ways can you get at least $7$ heads?

You flip a coin $10$ times. How many ways can you get at least $7$ heads? My answer. $$\binom{10}{10}+ \binom{10}9\cdot\binom{10}1 + \binom{10}8\cdot\binom{10}2+\binom{10}7\cdot\binom{10}3$$ You ...
4
votes
4answers
188 views

There are 10 marbles in a bag. $6$ are red and $4$ are blue. You must chose at least 1 red marble. In how many ways can you chose three total marbles.

I thought the answer is $^9C_2$ since the first (red) marble didn't count. You have to pick a red marble which reduces the total count from 10 to 9. The answer is 116 possible ways.
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0answers
30 views

Ways to fill matrix such that permanent is zero [duplicate]

Given a matrix $A$ of size $N \times N$ consisting of all zeroes. Find the number of ways to fill the matrix with $N + 2$ ones such that the permanent of matrix is zero. The permanent of a matrix is ...
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1answer
39 views

Possible paths between two points A and B

Assume that the following figure represents a map of part of a city very well traced, where the lines are streets. A person is at point A and you want move to point B, but the condition is that you ...
7
votes
1answer
52 views

The game of balls

$6$ balls marked as $1,2,3,4,5$ and $6$ are kept in a box. Two players A and B start to take out $1$ ball at a time from the box one after another without replacing the ball till the game is over. The ...
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votes
3answers
40 views

Combinatorics: Distinguishable and Indistinguishable Variables

$3$ men and $5$ women (each of the $8$ being different from all the rest) are lined up for a photograph. Also in the line are $3$ identical armadillos which are completely indistinguishable from each ...
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0answers
20 views

From How many different ways can we form subsets of $X∪Y$ with $r + s$ elements if $r$ of its elements must come from $X$ and $s$ of $Y$?

Consider two sets $X$ and $Y$ disjoint with $n$ and $m$ elements respectively. From How many different ways can we form subsets of $X∪Y$ with $r + s$ elements if $r$ of its elements must come from $X$ ...
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4answers
69 views

Why is $\binom{n}{2} = \sum_{i=0}^{n} i = \frac{n(n+1)}{2}$? [duplicate]

I can see this easily by definition. But will it have combinatorial meaning of $\binom{n}{2} = \sum_{k=1} ^{n-1} (k)?$
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4answers
67 views

Probability of creaating“MISSISSIPPI”

The original problem is: "You choose a letter at random from the word Mississippi eleven times without replacement. What is the probability that you can form the word Mississippi with the eleven ...
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1answer
36 views

How do I solve this problem using Permutation and Combination? [closed]

The sum of proper divisors of 72 (1 and 72 excluded) is i. 195 ii.122 iii.194 iv. None of these I have already solved it by adding the divisors (which was easy to do and the only approach I could ...
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1answer
32 views

Equation for real number to binary conversion

A real number can be converted into binary as given in the attached figure. How can we represent this process using an equation? The method of converting a real number to binary
3
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4answers
68 views

A student is given an exam consisting of 8 essay questions divided into 4 groups of 2 questions each.

From ETS Major Field Test in Mathematics A student is given an exam consisting of 8 essay questions divided into 4 groups of 2 questions each. The student is required to select a set of 6 ...
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0answers
20 views

What is the equation to determine how many combinations can be found when there are two sets with three letters and each can have one of two colors.

I have two sets (set A, set B) of three letters (X, Y, D). Each letter can have one of two colors (red or blue). How many different combinations are possible. I would like the equation to determine ...
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1answer
42 views

number of possibilities for a 6 digit number with contraints

We have a 6 digit number _ _ _ _ _ _ ...
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0answers
16 views

How to calculate upper bound of combinations?

I have 10 nodes and 14 RF antennas and 14 millimeter wave antennas. How can I calculate the upper bound of all combinations of distributions of these RF and millimeter wave antennas on the 10 nodes ...
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3answers
245 views

Number of surjective functions from a set with $m$ elements onto a set with $n$ elements

I was trying to calculate the number of surjective (onto) functions from A to B. Let a set $A$ contain $m$ elements and another set $B$ contain $n$ element i.e. $$|A|=m, \quad |B|=n.$$ Now, if $n>m$...
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0answers
43 views

Calculate combinations

I've developed a PHP script that calculates the sum of a series of numbers in various combinations and compares the result with a given total value. This is already working, but the problem is that ...
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2answers
34 views

How many combinations? [closed]

How many sequences of 3 characters can be made from 100,000 characters? The same character can appear once, twice or three times. For instance 1,1,1 or 1,1,2 or 1,2,2. And its the sequence that is ...
2
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1answer
57 views

Number of ways of build a binary matrix with constraints

We have an $N\times N$ matrix U whose elements can only be $0$ or $1$. I would like to count the multiplicity of the following matrix: $E$ non-zero entries $\sum_i U_{i j} = a_j$ i.e. fixed sum ...
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1answer
35 views

Combinations Of Several Trials

Let '$n$' be the number of trials. Let '$o$' be the number of outcomes for each trial. Let '$a$' be the first outcome for each trial, let '$b$' be the second, '$c$' the third, etc. How many ...
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0answers
32 views

New Twist - sum of numbers printed on billiard balls in a bag

Given that all the balls in a bag have one number (either 0, 1, 2, 3, or 4) printed on them, and given that there are 30 balls in the bag, how many possible combinations of balls can be taken from the ...
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2answers
39 views

Calculating the Search Space for Dr. Eureka's Puzzle

I have to prepare an algorithm to solve the puzzle part of Dr. Eureka, a multiplayer game from Blue Orange Games. This is part of a research project that also involves computer vision and robotics. ...
2
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1answer
29 views

Reversing cyclic

P(A) is a cyclic permutation graph A if each vertex is shifted by one in P : (P({1, 2, 3, ..., n - 1, n}) = {2, 3, ..., n - 1, n, 1}). e.g : cyclic permutation of 2 - 4 - 1 - 3 is 3 - 1 - 2 - 4 ...
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0answers
19 views

Simplification of Combination

I want to simplify this formula : ${n-1 \choose k-1}$ - ${l \choose k-1}$ - ${r \choose k-1}$ Where $l + r + 1 = n$ I tried to solve it using ${n!\over (n-k)!k!}$ , but wasn't able to get any ...
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0answers
34 views

Number of ways to form distinctive number given sum of digits

We are given sum of digits M and maximum number of decimal places N. We have to find the count of number that can be formed such the the sum of digits is M and the consecutive digit of the number ...
2
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4answers
84 views

How many fair dice of this kind exist?

I am not talking about the shape of the dice here, I am talking about another type. You will see what I mean soon. For example, when there are 1 dice, a normal dice is a fair dice, because the ...
1
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1answer
41 views

Set Partition of N elements into K groups. All equal elements.

I am trying to get the total number of possible set partitions of N elements into K groups, in a case where all the elements are the same. For example, in the case of $N=2$ and $K=3$ the possible ...
0
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1answer
39 views

Longest possible subsequence present with a given condition

Let a Domino represent an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the ...
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votes
1answer
64 views

Sum of Combination. Calculate $\sum_{k=0}^{10} \binom{10}{k}^2$ [closed]

I wanna know the answer of this problem and solve it: $$\binom{10}{0}^2+\binom{10}{1}^2+\binom{10}{2}^2+\cdots+\binom{10}{10}^2 = \ ?$$ Can you help me? Thanks in advance.