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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In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together?

In how many ways can $5$ students and $3$ teacher sit around a table so that no two teachers are together? My attempt: $5$ student can sit $(5-1)!$ in round table. A teacher can sit between ...
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1answer
28 views

Infinite Marbles in a Jar with Known Distribution

Let's say I have infinite number of marbles in a jar and $90\%$ of them are red and $10\%$ are green. If I pick $25$ out of the jars (with or without replacement probably doesn't matter because the ...
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Comparing the task complexity of installing three different offenses for American style football in three days

I want to identify the inherent difficulty of installing three separate American rules football offenses by their complexity of practice schedules in three days then relate those offenses back to one ...
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3answers
231 views

How many 6 digit numbers are possible with no digit appearing more than thrice?

How many 6 digit numbers are possible with at most three digits repeated? My attempt: The possibilities are: A)(3,2,1) One set of three repeated digit, another set of two repeated digit and ...
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1answer
28 views

the MISSISSIPPI problem - 5 letter word and 6 letter word with constraints

1) Number of ways of selecting 5 letters such that 3 are alike of one type and 2 are alike of another type. There are only 2 ways of selecting 3 letters of the same type i.e., III or SSS. Now with ...
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1answer
25 views

Combinatorics [sandwich toppings]

I'm trying to figure out how to solve this excercise; You can order a sandwich from 5 different types of bread, you can have butter, lettuce or neither. You get to choose from 3 types of meat, and ...
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1answer
26 views

How do we solve these permutation and combination questions? [on hold]

Q1 In how many ways a panel of six doctors is selected from five surgeons and six physicians if condition is surgeons are more than physicians. A 82 B 81 C 65 D 135 Q2 Find the no. of ...
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1answer
25 views

counting the forecasts of 20 chess games

I have a Question... The results of 20 chess games (win, lose, draw) have to be predicted. How many different forecasts can contain exactly 15 correct results? I don't really understand this ...
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71 views

Optimizing Overwatch Team Composition by Player Hero Preference [on hold]

I am wordy by nature - my apologies. My attempt at a TL;DR - I want to design a small tool that optimizes the team composition of a video game based on minimizing the sum of provided player ...
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2answers
37 views

calculating probability of a kid have a sister

Suppose there is equal probability of Boy (B) and Girl (G). If a family has 3 kids, wondering what is the probability of a kid has a sister in the same family. I have two methods of calculation, ...
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35 views

Need Help - Plane flight Probability Combinations [on hold]

The flight time of an aircraft depends on its velocity relative to the ground. This speed depends she even own the speed of the aircraft in an inertial frame that moves at speed constant which is the ...
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Awsome problem of combinations - [on hold]

Among the travelers arriving at Montreal airport to board a plane, (100 * p1)% arrive at destination while the others are in transit. Among those who arrive at their destination, 4% started their ...
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100 shoelaces, pick 2 random ends and tie them together, what is the probability that a loop is created?

The question is: There are 100 shoelaces in a box. You pick two random ends and tie them together. Either this results in a longer shoelace (if the two ends came from different pieces), or it ...
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0answers
30 views

Any optimized algorithm to find all k values where the sum of k values = S with k>=2 in a defined list [1,2,3,4,…n]? [on hold]

exemple to sole with k=3 list = 1,2,3,4,5,6,7,8,9,10, .. N Sum (S) = 13 Results : [1,2,10],[2,5,6], ..
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0answers
16 views

Evaluation sum indexed by non decreasing sequences

During solving a problem from probability theory, I've met the following sum to evaluate: $$p_n(N) = \frac{1}{N!}\sum_{0\leqslant k_1\leqslant\ldots\leqslant k_n\leqslant N}\frac{k_1\cdot\ldots\cdot ...
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1answer
52 views

What are some efficient ways to go about a problem where you cannot exceed the other by 2?

I need an efficient way to go about this problem, for practice for my problem solving test. This is not a part of the actual test. This is the type of question that I am struggling with There are two ...
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2answers
36 views

COMBINATORY LOGIC: Cards extraction from a deck of 32 cards.

5 cards are extracted simultaneously from a standard deck of 32 cards (8 cards for each of the four suits (hearts, diamonds, spades and clubs): 7,8,9,10, Jack, Queen, King, Ace). How many ...
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1answer
16 views

Odds of an event happening

Trying to get my head around the correct way of approach this. You are able to use any letter of the alphabet or number allowing for 36 options, with this you are to create PIN of length 4, for ...
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3answers
22 views

How many duplicates are eliminated via sorting?

This is a problem I've encountered when using a set as a key for a lookup table. Say I have to map a set of letters (lets say, $4$) to some (unspecified) result with a dictionary/array/etc. For ...
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1answer
21 views

Hints for Solving Elementary Combination problem of Doughnuts. [closed]

There are eight varieties of Doughnuts, if a box contains $1$ dozen doughnuts how many different option are there for a box of doughnuts ?
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1answer
24 views

Selects a number from the set $\{−10,−9,−8, …8, 9, 10\}$

Any help with this problem is highly appreciated. This shouldn't be a very difficult question, but I am confused with $f(x^2)$. Consider a function $f(x)$ that has zeroes $4$ and $9$. Given that ...
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Number of SDRs of ${A_1,\dots,A_k}$ whose minimum size of $A_i$ is $k$ [closed]

Suppose that $A_1, A_2, \dots,A_n$ refer to a set system, which the size of minimum of them is $k$ (for any $i$ such that $|A_i|≥k$) this family (set system) has a system of distinct representatives. ...
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A SDR extension problem [closed]

enter image description here I think we should use induction. for subset I when $|I|=1$ then we have for any $i\in \{1,\dots,n\}$ $|A_i|\leq 2$. Then for any $A_i$ we have $2$ SDR. then we suppose ...
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1answer
41 views

How many coefficients in $(x_1 +x_2 + \cdots + x_L)^N$?

How many coefficients in $(x_1 + x_2 + \cdots + x_L)^N$? That is to say, what is the number of coefficients when it represents as sum of products.
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62 views

Sequence of integers in given range that sums up to given value

I'm trying to find out, if there is a way to find the total number of possible combinations of integers $x_i \in [l,u] \cap \mathbb{Z}$ for all $i = 1,\ldots,n$ that sum up to $A$. Generally, ...
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1answer
24 views

Two Dimensional Choose or Combination Problem in Statistics

I have a two-dimensional choose problem. ...
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1answer
18 views

Total number of integral solutions to the factors of a given numbet

Let $a$ be a factor of $120$ then what are the total number of positive integral solutions to $xyz=a$ including 120. The answer is $320$ . After wasting almost $15$ mins in getting the factors of each ...
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2answers
110 views

Example in Combination, is there any solution?!

Is there any idea to solve such a question? I have $40$ pens that includes $20$ white pens and $20$ black pens, I decide to distribute these pens among $4$ students that every student gets at least ...
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0answers
24 views

Number of ways to allocate balls

We are given $N$ buckets and $X$ ways to allocate balls in each possible pair of buckets. How many ways of distributing balls in all buckets exist ? For example - If we have $3$ buckets and $7$ ways ...
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2answers
54 views

How many ways to pick $25$ objects from $5$ types with at most $10$ of any type?

How many ways to pick $25$ objects from $5$ types with at most $10$ of any type? The answer is: $C(25+5-1,25) - 5C(14+5-1,14) + C(5,2)C(3+5-1,3)$ Where do the numbers in this answer come from? I ...
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Counting Principle - how many different ways to insert a slide into a projector

Can anyone help me on this? It is a math contest problem. To solve the problem, I checked how a projector works online. But still can't figure out how to solve the problem. Thank you very much! ...
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2answers
34 views

Number of possible team combinations

The baseball team is made up of 7 girls and 8 boys. There are 9 people on the field at the same time in a baseball game. If at least four of the people on the field must be girls, how many different ...
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0answers
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Coin Combinatorics: Minimum amount of coins for given value

I was just doing a bit of thinking on coin combinatorics today when I began thinking of this question: You have a set of coins, greater than £2, but you cannot pay someone £2 without needing change. ...
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Number of quadrilaterals formed out of N points.

How can I calculate the number of $4$-vertex convex polygons (quadrilaterals) that can be formed out of $n$ given points, where $n \geq 4$? Note: Points can be collinear. So triangles with $3$ sides ...
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32 views

Number of ways to represent a number as a sum of K numbers in subset S

Let the set $S = \{ 1 , 2 , 4 , 5 , 10\}$ Now I want to find the number of ways to represent $x$ as sum of $k$ numbers of the set $S$ (a number can be included any number of times). If $x = 10$ and ...
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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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1answer
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A combinatorial problem - special convex polygon

I was trying to solve this question, yet I could not come up with a straightforward proof... it says given a set S containing 100 points on a plane, no 3 on a line, there is a convex polygon whose ...
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1answer
108 views

Why is a Pair of Tens better than a Pair of Aces in Texas Hold 'Em?

A couple years ago I developed a program to calculate the optimum betting amount for a round of Texas Hold 'em by using the Kelly criterion. In the process of computing the probability of winning for ...
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how to get sum x by adding exactly k elements of array a with repeatation but unique pair

i have an array of positive integers and two numbers $x$ and $k$. I have to calculate the number of unique possible combinations such that the sum of $k$ elements from array (repetition allowed) is ...
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3answers
703 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
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2answers
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Value of n summations of 1 $\sum_{0 \le a_1\lt a_2 \lt a_3 …\lt a_k \le n}1$

I need to find $$\sum_{0 \le a_1\lt a_2 \lt a_3 ...\lt a_k \le n}1$$ My attempt: I think it is equal to $ ^nC_k $ as $$\sum_1^n{1} = n = ^nC_1$$ $${\sum \sum }_{0\le i \lt j \le n} 1 = \frac ...
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1answer
40 views

$48$ balls, draw $6$ balls, how many bets are needed to ensure that exactly $1$ ball matches in at least $1$ bet?

I have a method to calculate it. But it does not make sense. So most likely I misinterpret the numbers. Please show me from where it goes south. The rule This is a simplified version of the lottery ...
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1answer
33 views

Permutation and Combination to find pairs

In how many different ways students can be paired such that no pair consists of 2 boys. Given :- Total students = 10, Girls = 7, boys = 3. What my approach is 3 boys can be paired with 7 girls like ...
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Strong induction postage stamp problem [closed]

Using 5 cent, 11 cent, and 17 cent stamps, what are all possible amounts of postage that can not be formed? Prove your answer. Part of your answer needs to use strong induction.
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How to use sub-pattern frequencies to calculate the authenticity the main pattern

I'm trying to determine the probability that something is real by comparing the frequency of the sub-permutations it contains. Here is an example: You want to guess whether or not a random symbol is ...
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3answers
62 views

Five people have applied for three different positions in a store. In how many ways can the positions be filled?

Five people have applied for three different positions in a store. If each person is qualified for each position, in how many ways can the positions be filled? Can someone tell me if I have to ...
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2answers
31 views

Juice problem combinations

I have no idea how to start this question... It's impossible in my mind. Help please The owner of a convenience store reports that 500 people who bought bottled fruit juice in a recent week. 400 ...
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29 views

How to determine a matrix when the sums of rows and columns are given

I was given a $5\times30$ matrix with each element of the matrix can only take 0 or 1. It is given that the sum of each row is 8 and the sum of the first 20 columns equals to 1 while the sum of each ...
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1answer
33 views

Combinations of up to $n$ out of $m$ elements

Given a set of $m$ unique elements to choose from, and using at most $n$ elements in a combination, how many combinations can I have? A combination can repeat an element more than once, as long as the ...
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How to calculate - Combinations of n items that are at least 50% differentiated?

If I have a set of 10 items, how do I create combinations of that set that are at least 50% different from every other possible combination? Any way to do this?