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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Combinatorics Question with vowels

How many 8-letter words contain exactly 5 vowels (a,e,i,o,u)? What if repeated letters were not allowed? This question has two parts to be answered. The first part is,"How many 8-letter words ...
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Calculation of finding way to make a password

A computer password requires you to use exactly 1 uppercase, exactly 3 lowercases, 3 digits and 2 special charecters(given that there are 33 special charecters that can be used)(every thing can be ...
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How many variatons of winners are there in 15 1vs1 matchs

I am trying to find a program or some way to show me every variation of winners from just 5 matchs of 1vs1. I think there should be 900 variation i just need a way to see it all written down .. For ...
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If Anna goes from point A to point B, each step can only move up or move right. How many method(s) is / are there?(reference the grid below)

If Anna goes from point A to point B, each step can only move up or move right. How many method(s) is / are there?(reference the grid below) I’ve just recently learned permutations and combinations. ...
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3answers
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Where is my error in this combinations word problem?

A student has to answer $10$ out of $13$ questions in an examination. The number of ways in which he can answer if he must answer at least $3$ of the first five questions is: The answer is not $^5C_3 ...
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Number of substrings possible with even characters

Consider a string 'ABBAA' Possible substrings with even number of characters are 4 'ABBA' : Count of 'A' is even and 'B' is even 'AA' : Count of 'A' is even and 'B' is even ($0$) Similarly 'BB' and '...
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Variations of the problem of the number of ways to divide $N$ items into $K$ bins

I can use the stars and bars approach to solve the problem of dividing $N$ items into $K$ distinct bins with no restrict on the number of items each bin has to have. It's simply $$ \binom{N+K - 1}{K - ...
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2answers
76 views

Counting squares in a part of a graph. What is the correct answer?

Consider the following set of points in the $x - y$ plane: $$ A = \{(a,\ b)\ |\ a,\ b\in \mathbb{Z}, \ 1\leqslant a\leqslant9\quad \text{and} \quad 1\leqslant b\leqslant 5\}. $$ Find the number of ...
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Sort a Bernoulli r.d. sequence with different success probabilities such that in expectation the maximum consecutive success events is maximized

For example, if the realization of this sequence is {0011001110}, then the maximum consecutive success events is 3. If the realization of this sequence is {0011110111}, then the maximum consecutive ...
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Calculate all possible combinations of 1, 2 & 3 which equal 67

My son (a third grader) was given this problem to solve. Is states that: In a basketball game, one of the teams scored 67 points. If baskets can be worth 1 point for a free throw, 2 points for a field ...
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1answer
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2 numbers are chosen from $1,3,5..151$ and multiplied in all possible ways. Find number of ways such that product is a multiple of 5

Number of available elements = 76 Elements that are a multiple of 5 = 15 Now one perfectly acceptable answer for this is $$\binom {30}{2} +\binom{15}{1}\times \binom {46}{1}$$ I understood how it came,...
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1answer
28 views

Find the total number of positive integral solutions for $x,y,z$ such that $xyz=24$

I recognize that this is related to the stars and bars algorithm, but I am just not able to apply it here The general method is factor the number, in this case is $2^3\times 3$, but what next? It’s ...
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2answers
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Why is 23 this summation's upper limit?

Can someone explain why $12$ and $23$ are the lower and upper limits for the series equation in the answer to this question: BlackJack Card Probability when Counting Cards
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1answer
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The dollar menu at your favorite tax-free fast food restaurant has 10 items. You have $15 to spend.

Full Question: The dollar menu at your favorite tax-free fast food restaurant has 10 items. You have $15 to spend. How many different meals can you buy if you spend all your money and: a. Purchase at ...
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General solution for a cascading recurrence relation

My main question: Starting from a simple recurrence relation: \begin{equation} Y_i = aY_{i-1}+(1-a)X_i\tag{1} \end{equation} I can easily find that the general solution, if $Y_0=0$, is: \begin{...
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2answers
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Consider functions f:{1,2,3,4,5}→{a,b,c,d,e,f}. How many functions have the property that f(1)≠c or f(2)≠f, or both?

So I'm working through this problem, and I think my error is that I'm over counting, but I'm not sure. Question: Consider functions f:{1,2,3,4,5}→{a,b,c,d,e,f}. How many functions have the property ...
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You are tasked with putting the 27 identical dodgeballs away into 4 bins. This time, no bin can hold more than 7 balls [closed]

Question: After another gym class you are tasked with putting the 27 identical dodgeballs away into 4 bins. This time, no bin can hold more than 7 balls. How many ways can you clean up? So I believe ...
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31 views

Establish formula for a combinatoric problem

This is a hard problem. Please read the following. There can be multiple farms and multiple farmers in a problem. Each farmer can attend to either one farm or two adjacent farms at a time- whether a ...
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Math behind this SQL problem

I have the following 'sorted by row' lists (2nd column), in which every row produces an output (3rd column, and 4th column). This output has been found without using formulas and it represents a ...
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1answer
44 views

How to distribute 10 distinct envelopes into 3 (identical) mailboxes?

As in the title, I would like to distribute 10 distinct envelopes into 3 mailboxes. Let us also assume that some mailboxes can remain empty. If they were identical envelopes, I know that we can use ...
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3answers
94 views

Number of combinations where a, b, c, and d appear?

I have 4 "containers", each with elements a, b, c, d, e. Each step, I randomly choose one element from each container and write them together. For example, I could have an outcome of "...
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What are the combinations that can be generated from a three digit code with 4 different numbers? [closed]

The numbers are 3, 6, 7, and 9. How many combos in threes can be made without repetition?
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1answer
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Can these connections be considered as permutations or combinations?

Any help would be highly appreciated.
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Why is this problem a question on the topic of permutations/combinations?

I am reviewing some random problems about combinations/permutations. I stumbled upon the following: In a group of $20$ people, how long will it take each person to shake hands with each of the other ...
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1answer
59 views

How many arrangements of red and blue balls are there so that, the number of red balls with: the ball immediately to the right is also red, is $9$.

The question is too long to fit in the title, but I tried. $50$ balls: $23$ indistinguishable red balls; $27$ indistinguishable blue balls. The balls are arranged in a line. How many distinct ...
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3answers
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Combo problem, complementary counting

You have 5 blue nails in a column and 3 red nails in another column. You can attach a string between any red nail and blue nail. How many ways can we attach these strings such that every nail has at ...
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0answers
19 views

Is this way correct to solve this question in Permutations and Combinations?

I have this question in Permutations and Combinations, You have $3$ sections $A, B$ and $C$. You need to answer $12$ questions out of $20$ in section $A$. $4$ out of $6$ from section $B$. In section $...
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4answers
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Probability of 4 colors sequence with repetition (8 choices of color)

Knowing that the sequence contains exactly two colors, what is the probability that one of the two colors will be repeated exactly three times? A = The probability that one of the two colors will be ...
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3answers
35 views

Probability question about putting X items in K things

The question is: "8 employees want to book meetings with their manager on randomly chosen days of the work week, from Monday till Friday. What's the probability that the manager has at least one ...
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1answer
31 views

Find the number of positive integer pairs $(x,y)$ such that $\frac 1x + \frac 1y=\frac{1}{2007}$ such that $x<y$

$$2007(x+y) =xy$$ $$\implies (x-2007)(y-2007)=2007^2$$ Now the values so $x$ and $y$ should depend on the factors of $2007^2$, which is something I am not very sure about How should I proceed?
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Find all possibilities to split a set of non-necessarily disjoint elements (prime factors for example) into two subsets.

To find all possibilities of splitting a set of non-necessarily disjoint elements, let's take the $8$ prime factors $e~e~f~g~h~h~h~h$ for example, into two subsets: How many possibilities exist? (...
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1answer
29 views

Error in two approaches when finding the number of distinct integers between 100 and 1000 [closed]

My first approach there are 3 places A B C to be filled. For the unit place C :- There are 10 possible ways to fill it because possible outcomes equals 0 to 9. For the tens place B :- In order to be ...
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How do you describe logically in words that there are 2 paired selections for picking two different items simultaneously?

I have just written an article about probability and combinations, but am having trouble describing in words the logic behind simultaneously choosing one each of two different items. I would be ...
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Find a function that maximize the number of tokens in my buckets with following constraints [closed]

I need help with the following problem. Let 'k' be the possible colors a token can be Let 'm' be the number of empty buckets available Let n=2 be the number of maximum tokens per bucket. Only tokens ...
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1answer
22 views

I want to obtain partition of an integer with an initial value and

I want to obtain a partition of an Integer with an initial value and the value following it is smaller and the value following it is smaller than the previous value and no value repeats itself. within ...
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2answers
23 views

Number of codes with $5$ distinct characters using $5$ letters and $9$ numbers

The letters; $A, B, C, D, E$ and the number; $1, 2, 3, 4, 5, 6, 7, 8, 9$ are to be used to form codes with five distinct characters. Each code has at least one letter and has at least one number. How ...
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2answers
52 views

Find all the possible numbers whose sum is x but … [closed]

I want to find all the number whose sum is x but but series of number start with a specific number and next number is always smaller then the number before it For example I want 41 but first number in ...
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1answer
42 views

Possible combinations of elements from different sets

I have the following problem. Let's start from a simple case. Suppose i have to sets, A and B with 4 and 5 elements respectively. I need to pick a certain number of elements from these two sets that ...
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2answers
100 views

Balls and bins - picking drawing one ball from each bin and calculating probability that all balls are black

Assume we have $3$ bins: bins $L_1, L_2, L_3$, in each bin, there are $3$ balls. Alice is picking one ball from each bin at random. The balls can be either white or black. Alice does not know how many ...
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4answers
55 views

Counting permutation of duplicate items

I am not really sure I fully understand the formula for finding the number of permutation of duplicate items. The formula is: $$\frac{n!}{p!q!r!\cdots}$$ Where do the factorials in the denominator ...
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0answers
23 views

Combinations of 4 variables with 3 values each [closed]

I'm pretty new to Excel and am trying to find a way to automatically generate all possible combinations (order not important) of 4 variables (A, B, C, D) with 3 values (1, 2, 3) each. So there are ...
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2answers
31 views

Combinations of three mutually perpendicular edges of a cube. [closed]

I need help in determining the possible combinations of mutually perpendicular edges of a cube, taken three at a time. Two examples are shown in the image attached. enter image description here
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1answer
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Rank of a permutation given k things are arranged out of n things

There are some standard dictionary problems like given a word bottle, one can find the rank of this word alphabetic order by making sperate cases, but my question is that if you can choose any three ...
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1answer
35 views

Finding combinations of 81 discrete values that add up to a known value

First, I'm a mechanical engineer, and not a programmer. So for practical purposes, the answer may be as simple as "Excel can't do that without VB", or even "Excel can't do that". ...
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0answers
22 views

square od combinatprial in discrete math [duplicate]

I have the following question: I am trying to prove that $\sum_{k=0}^{n} {\binom{N}{k}}^2 = \binom{2N}{k}$. I tried proving by induction $\sum_{k=0}^{n} {\binom{N}{k}}^2$, so I substituted k = 0 , 1,2,...
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1answer
39 views

probability: choose 2r shoes from n pairs of shoes

A closet contains n pairs of shoes. If 2r shoes are chosen at random (2r<n), then we know that there are $2^{2r} {n\choose 2r}$ different cases in the event "there are no matching pairs in our ...
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29 views

Question about intuition of calculating probabilities with combinations and counting methods(e.g. probability of a full house)

This is just a question to make sure I understand the intuition of using combinations to calculate probabilities from examples such as a deck of cards. Take for example finding the probability of a ...
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2answers
33 views

A rational number is given in the form:

A rational number is given in the form: $\frac{p}{q}$; $\frac{p}{q} \in (0,1)$ and $p$ and $q$ are coprime to each other. If $pq= 10!$ , what would be the number of ordered pairs $(p,q)$? A beginning ...
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2answers
29 views

Probability of drawing a white balls from the urn having identical white and identical black balls?

In an urn there are $m$ white balls and $n$ black balls. If a ball is drawn from the urn, then what is the probability that the drawn ball is white, given the condition that all white balls are ...
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2answers
61 views

Finding All Combinations

Question Consider the following prices on balls: \begin{alignat}{2} &\text{Red (R):} & 15&\\ &\text{Green (B):} & 6&\\ &\text{Blue (G):} &\qquad ...

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