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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Basic Probability using combinations [on hold]

(a) A committee of 5 people is to be chosen from a group of 10 (6 men and 4 women) (i) How many committees of 5 members can be chosen from 10 people? (ii) How many of the committees from (a) will ...
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3answers
19 views

How many 3 character combinations can be made using letters AND numbers?

I'm wondering how many possible 3 character combinations can be made using the 26 letters of the alphabet, and 0-9. I've seen that with just the alphabet you can create somewhere around 17k different ...
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0answers
23 views

How to calculate combinations by drawing out the spaces?

I'm learning about probability on khanacademy. They teach a certain method (they draw out the spaces) to calculate combinations. Two Examples: 1. Take the question "What is the probability to get ...
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3answers
76 views

Why does $n \choose r$ where $r = 1,n$ track $2^n$?

I bashed together a clunky ruby script to find the sum total of $n \choose r$ where $r = 1,n$ I wanted to determine how many lines of output I could expect from a script that produces all possible ...
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0answers
27 views

Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen

Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated ...
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1answer
31 views

Number of ways to invite people to dinner

So I have this maths contest problem which goes like this: Alfred has seven friends (we'll call them A,B,C,D,E,F and G). Each night for a week/7 days he can invite any group of three friends over to ...
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2answers
46 views

How do I evaluate this combinatorically?

I recently came across this problem and couldn't even start on it. Would someone be able to help me? Given $m$ identical symbols, say H's, show that the number ...
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1answer
37 views

What 12 numbers in a C(12,2) and C(12,3) combination produce non-repeating sums?

this is my problem: I need 12 distinct numbers, could be something like (3,9,15,17,27,etc.) that when added up either in combinations of 2 or 3, the sums do not repeat. For example, if the numbers ...
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2answers
32 views

Reduce Combination Formula

Hey i have to write a code for this: You can refer here: Picking Same Color Probability For the entire question. $\Pr(Success)=$$\sum\limits_{k=1}^{\min(m,n)}\frac{{m\choose k}\cdot{nm-m\choose ...
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0answers
31 views

Simple string permutations question

How many sequences of 5 letters are there in which exactly two are vowels? My approach There are $5^2$ different permutations for 2 vowels and $\binom{5}{2}$ ways allocate them. There are $21^3$ ...
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2answers
38 views

Basic Password permutation question

I'm reading the problem from this stanford material (http://infolab.stanford.edu/~ullman/focs/ch04.pdf). Can you please help me understand this? Question: At Real Security, Inc., computer passwords ...
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1answer
22 views

Calculate Number of Unique Combination of Numbers [closed]

I want to find the total of possible unique combinations of numbers example: 1234 = 24 combinations 1233 = 12 combinations 55666 = 10 combinations can I use math formula to calculate this ? ...
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3answers
31 views

Combination Of Sets Question

I have a bit of an advanced combination problem that has left me stumped for a few days. Essentially my question is if you have n sets of items, and you can select a different number of items from ...
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1answer
30 views

Multi stage probability events [on hold]

Three students are selected at random from a group of $6$ boys and $4$ girls. How many combinations are possible that contain exactly $2$ boys? The answer is $120$. I'm not sure where to begin.
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0answers
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Combinations of 16 people in 8 chairs

I have 16 people, and 8 chairs. How many different combinations of the 16 people can fit in the 8 chairs (order is irrelevant).
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2answers
35 views

(Fast way to) Get a combination given its position in (reverse-)lexicographic order

This question is the inverse of the Fast way to get a position of combination (without repetitions). Given all $\left(\!\!\binom{n}{k}\!\!\right)$ combinations without repetitions (in either ...
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0answers
94 views

Puzzle - In how many pairings can 25 married couples dance when exactly 7 men dance with their own wives?

Each married couple as well as each dancing pair consists of a man and a woman. How many possible pairings are there? Here is the same question with a different amount of couples. I read the answers ...
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1answer
99 views

Minimization of a combinatorial function

The following gamma function depends on the overall sum of $x_n,x_j,x_k$ $$\gamma(X)=\sum_{x_n+x_j+x_k=X}\left [ \left ( \prod_{i=1}^{s}(x_{ni}-1)!C_i^{x_{ni}} \right )\times ...
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1answer
17 views

combinations of 2 sets of item [closed]

If I have 2 sets of costumes, each has 6 parts ( lets say, hats, shirt, jacket, pants, belt and shoes ) How many unique possibilities of costume I will get when I mix and match them?
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0answers
16 views

Tournament creation [closed]

I have a competition with 8 teams and 4 different events. The competition will last 2 days and each team will play each activity once each day, so theyll play the eventime 2 times by the end. I need ...
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2answers
44 views

How many positive, three digit integers contain atleast one 7?

This is the Question: How many positive, three digit integers contain atleast one 7? For these kind of questions I have always followed a technique of first taking care of the restriction provided in ...
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1answer
45 views

Breaks the product based on ${x_j}$?

Can anyone help me to break this product into the series based on ${x_j}$ ? $$ \prod_{i=1}^{K}(1-x_i)$$ I want to break it to some function as below: $$ \sum_j\Psi(x_j) $$ I saw something like ...
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1answer
39 views

Number of combinations of selecting $r$ numbers from first $n$ natural numbers of which exactly $m$ are consective.

Number of combinations of selecting $r$ numbers from first $n$ natural numbers of which exactly $m$ are consective. Say $g(n,r,m)$ is the number of such combinations. The two cases of $m=r$ and $m=1$ ...
2
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3answers
101 views

Fast way to get a position of combination (without repetitions)

This question has an inverse: (Fast way to) Get a combination given its position in (reverse-)lexicographic order What would be the most efficient way to ...
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1answer
33 views

Prove it by theory of combination

Consider a $100$-sided regular polygon. How can I choose three points from the $100$ points such that they will be the vertices of a right angled triangle?
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2answers
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To prove an identity in permutation and combination.

I am trying to prove the following identity: ${n \choose 0}$ + ${n \choose 1}$ + $\ldots$ + $\frac{1}{2}{n \choose n/2}$ = $2^{n-1}$ where $n$ is even I know that I have to use few relations like ...
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3answers
64 views

In how many ways can a natural number be written as a sum of $2$ natural numbers?

For example, $7=1+6,2+5,3+4$. Hence $7$ can be written as a sum of $2$ natural numbers in $3$ ways.
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12- In a standard deck of 52 cards, how many ways can you deal out 4 cards that are all black or all not face cards? [duplicate]

I did the sad mistake of taking math in summer school to boost my average. I am stuck on a few questions. In a standard deck of $52$ cards, how many ways can you deal out $4$ cards that are all ...
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1answer
44 views

Number of ways arranging entries of a tuple - combinations or permutations

Let $x=(x_1,x_2,\ldots,x_n)$ be an $n$-tuple where $n$ is even In how many ways we can arrange such that exactly half of the entries are even ? My attempt is : As we are talking about possible ...
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1answer
33 views

Finding the number of combinations

A teacher distributes 7 books to 7 children (each student a books), on the next day she collects the books back and redistributes in such a way that each students get a new book. In how many ways can ...
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2answers
32 views

Unique combinations from 7 items where repetition is allowed, and order doesn't matter

I am trying to calculate the number of unique combinations from a 7-element set where repetition is allowed and order doesn't matter. For example: Suppose $S = \{a, b, c, d, e, f, g\}$, and I want 3 ...
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1answer
43 views

Sum of all the numbers with the given numbers repeated

How to find the sum of all the numbers that can be formed using the digits 4,5,5,6,6,6 (This includes 4,5,6,45,46,54,55,....,666554). I knew that the answer is 39345806. I just need to know the method ...
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1answer
39 views

How many matches are played?

A tennis club has $10$ couples as members. They meet to organize a mixed double match. If each wife refuses to partner as well as oppose her husband in the match, then in how many different ways ...
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1answer
47 views

How many arrangements of 4 letters, with 3 of them being distinct, are there?

I read an example of the "Counting Principle" where we want to find the number of possible ways to rearrange 4 distinct letters chosen from the alphabet. The answer for this one makes sense. This is ...
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1answer
25 views

How to finding permutations where some elements repeat?

Sorry if my question is not mathematically correct. Please help me fix it if there is a better way to phrase it. So first of all, I know that if you have a list of numbers {1, 2, 3} then the number ...
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1answer
33 views

Number of ways of choosing at least $k$ objects out of $n$

Suppose you have three distinct items $a$, $b$, $c$. You want to find how many unique sets you can get by choosing at least one item. For example, $\{a\}$ would form a unique set, and $\{a, b\}$ would ...
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1answer
23 views

Calculating combinations without duplicate values

I have 128 chairs, and 256 people. How many different combinations of the 256 people can be sitting in the 128 chairs? Order doesn't matter, and obviously the same person can't be sitting in more ...
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1answer
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Calculate $\left(\begin{smallmatrix}n \\ r\end{smallmatrix}\right)/{k^n}$ for very large $n$

How to calculate large $ \frac{\left(\begin{matrix}n \\ r\end{matrix}\right)}{k^n}$, given very large $n$. Since n is large enough normal methods of calculating $ \left(\begin{matrix}n \\ ...
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2answers
37 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
0
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1answer
64 views

Powerset with constraints

I have two sets $NUMBERS$ and $LETTERS$ with: $ NUMBERS = \{1, 2, 3, 4, 5\} \\ LETTERS = \{ A, B, C, D, E\}$ No I want the power-set of my sets, i.e. the set of subsets of elements from both ...
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How to solve this combinations with repetitions problem using generating functions?

Find the number of solutions to : $$x_1 + x_2 + x_3 + x_4 + x_5 = 10$$ where none of the variables can be the number $3$. I can solve this with Inclusion-Exclusion Principle, but I really love ...
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1answer
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Concept of combinations and number theory.

The number of six digit numbers of the form ababab(in base ten)each of which is a product of exactly six distinct primes?
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2answers
120 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
4
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1answer
24 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
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1answer
43 views

How many permutations of the letters DANIEL do not begin with D or do not end with L?

How many permutations of the letters DANIEL do not begin with D or do not end with L? The correct answer is 696. This answer does not make sense as there are 120 (5!) ways the letters can be ...
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1answer
38 views

In how many ways can the committee be selected if the girls must include either Roberta or Priya but not both? [duplicate]

A committee of three boys and three girls is to be selected from a class of $14$ boys and $17$ girls. In how many ways can the committee be selected if the girls must include either Roberta or Priya ...
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2answers
36 views

Permutations and Combinations Tricky Question

In a photo there are three families (six Greens, four Browns, and seven Grays) arranged in a row. The Browns have had an argument so no Brown will stand next to another Brown. How many different ...
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2answers
74 views

How many ways are there of splitting twelve people into two groups of the same size?

Twelve people need to be split up into teams for a quiz. How many ways are there of splitting them into two groups of the same size? I did $12 C 6$, which gives $924$, however the answer is ...
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2answers
118 views

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet?

In how many ways can five different sweets be split amongst two people if each person must have at least one sweet? I tried $5 C 1 + 5 C 2 + 5 C 3 + 5 C 4 = 30$, however, the answer is $20$. Any ...
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3answers
38 views

Combinations and Double Factorials

In a village, there are 10 boys and 10 girls. The village matchmaker arranges all the marriages. In how many ways can she pair off the 20 children, if homosexual marriages (male-male or female-female) ...