Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
5 views

Generating combinations using a butterfly network

I'm using a butterfly network to generate a random combination of a bitstring of length $n$ and weight $w$. Let me clarify it with an example. Suppose I want a random bitstring of length 8 and Hamming ...
0
votes
1answer
34 views

I need help to this math problem?

I have a memory card game 4 x 4. I have a memory game with 16 boxes. On the game, they are 2 similar cards. In total 8 different couples. I should know how many combinations I have in total like: A ...
0
votes
0answers
5 views

No of ways of selection when at least some number is to be included.

A candidate has to select 7 subjects from 3 different groups A, B, C, which contain 4, 5, 6 subjects respectively. The number of different ways in which a candidate can make his selection if he has to ...
2
votes
1answer
45 views

5 digits numbers such that when the sum of digits divided by 4 leaves remainder 2.

How many 5 digits numbers such that when the sum of digit divided by 4 leaves remainder 2. Example:- Consider a 5 digit number- (x1,x2,x3,x4,x5) Then (x1+x2+x3+x4+x5)= must be of form(4n+2) I tried ...
-2
votes
0answers
15 views

No of ways of selecting k letters from a given set of letters

There are p letters 'a' , q letters 'b' and r letters 'c' , where p,q,r $\in$ I . Then number of ways of selecting k letters out of these if $ p\lt k\lt q\lt r$ is? My attempt: Let x no. of letter '...
1
vote
0answers
20 views

Intuition behind combinations

Example problem: Suppose you have to select 5 cards from a deck: what is the probability of getting 4 diamonds and 1 spade. The solution should be : $$\ \frac{\binom{13}{4} \binom{13}{1}}{{52}\...
1
vote
1answer
13 views

Permutation and Combination in probability question - Choose team members

I would like to have your help and explanation on following question. For an 8-a-side football match, a coach has to choose the team from a squad of 12 boys. Only three of them can play as a ...
0
votes
1answer
69 views

Find : $\int_0^{\pi}\frac{(1-\cos px)(1+\cos x)}{\sin x}dx$

How I can evaluate in closed form this trigonometry integral $$I_p=\int_0^{\pi}\frac{(1-\cos (px))(1+\cos x)}{\sin x}dx$$ , $p≥0$ positive integer Original question is find : $(p+2)I_{p+2}-2I_{p+1}...
0
votes
0answers
27 views

Likelihood of getting flush, straight, etc

On Planet X, cards can take on a numerical value from $1$ to $7$ (inclusive) and their suit can be either red or blue. In a game of poker, each player gets three cards. 1) What is the probability of ...
0
votes
0answers
23 views

the number of ways to change an element in the permutation cycles from outside those cycles

I've been thinking about this issue for a while! For the set of $n$ elements, consider that there is a permutation over the whole set $\{1,\dots, n\}$ where $n-k$ elements are fixed. A way of ...
-1
votes
2answers
44 views

Number of functions $f$ on $\{1,\cdots,7\}$ s.t. $f(f(x))$ is constant

Let $A = \{1,2,3,4,5,6,7\}$. Find the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Please help me out. I tried all sort of combinations but not reaching ...
0
votes
0answers
20 views

Problems on combination of binary sequences

Show that we cannot find 171 binary sequences (sequences of 0’s and 1’s), each of length 12 such that any two of them differ in at least four positions. I assume $S$ to be any set of binary sequences ...
0
votes
0answers
13 views

Permutation and combination when certain objects are alike.

What will be the number of permutations and combinations when m objects are to be taken from a group of n objects, having 'a' and 'b' number similar objects? Example: Find number of ways of ...
2
votes
3answers
42 views

Getting a wrong answer on evaluating permutations separately

Good Day! I was doing some combinatorics problems when I got stuck. The problem was: Suppose that a teacher selects 4 students from 5 boys and 4 girls. If at least one boy and one girl must be ...
0
votes
1answer
33 views

How does distinguishability of boxes change the number of ways to distribute n objects into separate boxes [duplicate]

I was going through problems related to distributing n distinguishable objects all into boxes, a in the first, b in the second, c in the third where a+b+c = n. The solutions I've seen usually comes to ...
1
vote
1answer
41 views

Finding the Elements from a list which average to a Known Number

I apologize as I am very new to this so my question may not be written very well/formally. What I am trying to do is find which numbers out of a known list would average to a given number. Say the ...
1
vote
0answers
31 views

combinations word problems math

There are two different problems I am stuck on. In how many ways can we divide ten candies of the same kind among four children? I haven't found a way to solve this one 2.In how many ways can we ...
0
votes
1answer
35 views

In how many ways can we select $5$ children from this group so that Mary and Jane are always in the selection?

In a group of $10$ children no two kids have the same name. We know that Mary and Jane are among these children. In how many ways can we select $5$ children from this group so that Mary and Jane are ...
0
votes
0answers
18 views

How to find quantity of cyclical sequences of ordered pairs with permutation between adjacent pairs defined

I have manually identified possible combinations for a small set I have been working with and I'm not sure if I have overlooked some valid options. I would like to be able to calculate the quantity of ...
1
vote
2answers
54 views

How do I solve this combinatorics problem with conditions?

I have $N$ lattice points which are arranged linearly and equally spaced. I want to make connections(say with some wire or thread) with each lattice site with another. The first one has $N-1$ ...
1
vote
1answer
24 views

$\sum\binom{n}{a_1}\prod\limits_{p=2}^{k}\binom{n-a_{p-1}}{a_p-a_{p-1}}=k!{n+1\brace k+1}$

We have $$\sum\limits_{0\leqslant a_1<a_2<\cdots<a_k<n}\binom{n}{a_1}\prod\limits_{p=2}^{k}\binom{n-a_{p-1}}{a_p-a_{p-1}}=k!{n+1\brace k+1}$$ where instead of $\{a_1,a_2,\cdots,a_k\}$ we ...
0
votes
3answers
32 views

In how many ways we can split set$ \{a_{1},..,a_{9}, b_{1},.., b_{9}, c_{1} ,.., c_{9}\}$ into 9 set of shape $\{ a_{i}, b_{j}, c_{k} \} $

In how many ways we can split set$ \{a_{1},..,a_{9}, b_{1},.., b_{9}, c_{1} ,.., c_{9}\}$ into 9 set of shape $\{ a_{i}, b_{j}, c_{k} \}$?
4
votes
1answer
49 views

Combinatorial proof $\sum_i^{\lfloor{n/2}\rfloor} (-1)^i {n-i\choose i} 2^{n-2i} = n+1$

Give a combinatorial proof (double counting) that $\sum_i^{\lfloor{n/2}\rfloor} (-1)^i {n-i\choose i} 2^{n-2i} = n+1$ There was a hint that maybe $n$ bit binary numbers without 01 may help. (eg. 1001,...
1
vote
0answers
25 views

Number of ways to put $n$ objects with $k$ distinguishable groups into sets of size $a$, $b$, and $c$ $(a+b+c = n)$

If I have $n$ objects in $k ≤ n$ distinguishable groups, for the case of $k = n$ (all objects distinguishable), I believe the number of ways to choose, without replacement, $a$ objects, then $b$ ...
1
vote
2answers
14 views

Subset of combinations in larger set

I am a biologist and not a real mathematician. Hence some of the answers featured here are sometimes too complicated. My question is: I have set of 8 genes named PBX1,ESX1,PIM1,HBB,HBG,BCL11A,KLF4,...
0
votes
2answers
44 views

In combination problems, when does distinguishability of objects matter?

If I have $n$ objects, and asked the number of ways to choose $k$ of them, the answer is $\binom{n}{k}$ (example problem 1). Similarly if I have $n$ objects and asked to put $a$ in one bin, $b$ in ...
0
votes
1answer
29 views

Lucky draw Permutation or Combination

I need help with a question, I have no idea where to start. If someone can help me solve it, and explain it at the same time that would be great. Here it is: In a lucky draw, there are 20 names in a ...
0
votes
1answer
31 views

How many bytes contain exactly two 1s?

I know the answer is $C(8,2)$, but I was confused as to why the answer wouldn't be $P(8,2)$? Doesn't the order of the 0s and 1s matter? For example: 10010000 is different than 10100000, so don't we ...
0
votes
2answers
13 views

Combination or Permutation math question help

I need help with a question. I have absolutely no idea how to do this, can someone please explain how to solve this problem: A committee of 8 workers is formed selecting from a group of 6 men and 5 ...
0
votes
1answer
23 views

Arranging Letters and Plus Signs

We have 4 spaces which should be filled with $1$ letter and $3$ plus signs, $2$ letters and $2$ plus signs, or with $3$ letters and $1$ plus sign. In any of these cases, letters can not be repeated ...
-1
votes
0answers
18 views

How many unique ways to put H rings of k colors of n numbers (of each color) from a pile of r rings on m fingers? [closed]

Is there a formula? e.g. Let’s say you have 100 rings: green: 10 blue: 30 yellow: 42 red: 11 black: 7 how many ways can you put 35 of them on 5 fingers? (order matters)
1
vote
1answer
30 views

Questions relating to choosing a j-tuple (j≤n) of positive integers whose sum is at most n.

Let j and n be positive integers, with j ≤ n. An experiment consists of choosing, at random, a j-tuple of positive integers whose sum is at most n. a) Find the size of the sample space. Hint: ...
1
vote
1answer
26 views

number of ways to order 5 english and 4 french hits

the question is as follows: suppose a dj has 5 english hits and 4 french hits. in how many ways can these songs be played if no two english hits should follow each other? in how many ways can these ...
1
vote
1answer
50 views

Number of ways of buying five apples from six types of apples if no more than two of each type are purchased

Ok, this might be a stupid question, but I need some clarification on when problems have a direct solution of doing $\binom{n}{r}$ and some when you need to do $\binom{n+r-1}{r}$. A grocer sells ...
0
votes
1answer
19 views

The probability that $3$ organizers win

A local fraternity is conducting a raffle where $55$ tickets are to be sold - one per customer. There are $3$ prizes to be awarded. If the $4$ organizers of the raffle each buy one ticket, what are ...
1
vote
1answer
50 views

Number of $7$-digit telephone numbers with non-decreasing digits? strictly increasing digits? [duplicate]

I'm really confused with the question below. A phone number is a 7-digit sequence that does not start with 0. (a) Call a phone number lucky if its digits are in non-decreasing order. For example, ...
0
votes
1answer
28 views

How to show that intermediate values of $\binom{n}{\lfloor n/2 \rfloor }$ are all integers that does not exceed $\binom{n}{\lfloor n/2 \rfloor }$

Let n be a positive integer, and assume that j is a positive integer not exceeding $n/2$. Show that in $n \cdot (n-1)\cdot (n-2)\dotsm(n-j+1)\,/\,j!$, if one alternates the multiplications and ...
0
votes
1answer
14 views

Method of selecting the optimal average combination of highest numbers?

I know of pairing functions, but my problem may involve three to five variables. When I have a few robots configured with certain parameters for movement, and I want to evaluate the robots based on ...
0
votes
2answers
19 views

How many different words can be formed in which neither two consonants nor two vowels can come next to each other? [closed]

There are 6 letters of which 3 are consonant and 3 are vowels. How many different words can be formed in which neither two consonants nor two vowels can come next to each other?
0
votes
1answer
18 views

How do use inclusion-exclusion to find the probability of encountering all n outcomes in m tries

This is a repost as my last post broke some rules. Assume that every time you buy a box of Wheaties, you receive one of the pictures of the n players on the New York Yankees. Over a period of ...
-2
votes
2answers
41 views
1
vote
1answer
22 views

Let $S=${$1$,$2$,..,$n$}.In how many ways can we choose two subsets $A$ and $B$ of $S$ so that $B \neq \emptyset$ and $B \subseteq A \subseteq S$?

Let $S=${$1$,$2$,..,$n$}.In how many ways can we choose two subsets $A$ and $B$ of $S$ so that $B \neq \emptyset$ and $B \subseteq A \subseteq S$? My first approach involves finding the sum $\sum_{k=...
0
votes
0answers
15 views

A question about contestants linked with combinations

Suppose A and B are two candidates contested in an election who secured 11 and 7 votes respectively. Then the number of ways if A stayed ahead of B throughout the counting process of votes is?
0
votes
0answers
15 views

Distribution in the combinatorial optimization

There is a set A containing $N$ elements. Then we randomly choose $M$ out of $N (N≥M)$ elements to form a subset B and get the objective function $f(B)$. Different $B$ may lead to different $f(B)$. ...
0
votes
1answer
20 views

Distribution of the knapsack problem

I am considering a special knapsack problem. The knapsack capacity is $M$. There are $N$ items ($N≥M$). The weight of each item is $1$. The profit for each item $i$ is $p(i) ≥ 0$. Thus, $M$ items can ...
2
votes
4answers
81 views

If a fair coin is flipped 5 times, what is the probability of getting exactly 3 heads

Q: "If a fair coin is flipped 5 times, what is the probability of getting exactly 3 heads?" A: we can do this question by drawing blanks strategy. for eg. $$\left(\dfrac{1}{2}\right)\left(\dfrac{1}{2}...
1
vote
1answer
24 views

Using Draw Blanks approach to solve a probability question of drawing 2 red, 2 blue and 1 green marbles.

Q: "There are 6 children at a family reunion, 3 boys and 3 girls. They will be lined up single-file for a photo, alternating genders. How many arrangements of the children are possible for this photo?"...
0
votes
3answers
36 views

arranging 2 blue balls, 2 red balls and 1 green ball [closed]

How many ways are there to arrange 2 blue balls, 2 red balls and 1 green ball? My Answer: $$\frac{5!}{2!*2!}$$ If this is incorrect then please help me understand where I am wrong. But if this is ...
3
votes
0answers
37 views

Dinner party courses needed to have everyone sit with everyone else (repeats allowed)

There are a large number of similar questions on this site, but most of them seem to have an additional constraint that I do not have. I am organizing a dinner party for $P$ people sitting at $T$ ...
0
votes
1answer
39 views

Prove your identity using a block-walking argument.

Prove that $\binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2=\binom{2n}{n}$ by using a block-walking argument. I found the identity but I wasn't able to find a block-walking argument. Could ...