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.

learn more… | top users | synonyms

2
votes
3answers
30 views

Explain solution to calculating number of ways of selecting 3 objects from 5 objects (repetitions permitted)

The solution is: Let $Y=\{y_1, y_2, y_3,y_4,y_5\}$ Then, each selection corresponds to a triple $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$. A bijection from this set of triples to ...
0
votes
1answer
23 views

Why does this combination correspond to an injection from $\mathbb{N_2} \rightarrow Y$?

Suppose 3 people each select a main dish from a menu of five items. How many distinct choices are possible if 2 people select the same dish? The solution: Let $X$ be the set of 3 people and $Y$ be ...
1
vote
3answers
94 views

How to deal with linear recurrence that it's characteristic polynomial has multiple roots?

example , $$ a_n=6a_{n-1}-9a_{n-2},a_0=0,a_1=1 $$ what is the $a_n$? In fact, I want to know there are any way to deal with this situation.
1
vote
4answers
71 views

Probability of drawing at least 1 red, 1 blue, 1 green, 1 white, 1 black, and 1 grey when drawing 8 balls from a pool of 30?

Given a pool of 30 balls (5 of each color). When drawing 8 balls without replacement, what is the probability of getting at least one of each color? Related: Probability of drawing at least one red ...
0
votes
1answer
18 views

How Many Unique Character String Can Be Made From 62 Characters?

I'm working on a programming algorithm and need a little math help. I'm in 10th grade and I think the question I'm asking is actually a permutation and combination logic question. Okay, so I've 62 ...
1
vote
2answers
33 views

Probability of drawing at least one red and at least one green ball.

Given a bag containing 10 red balls and 10 green balls. When you draw 6 balls, without replacement, what is the probability that you will have at least 1 red and 1 green ball. I attempted this ...
5
votes
3answers
151 views

How many ways can 5 dice produce a total of 20?

How many ways can $5$ dice produce a total of $20$? I set up the equation $x_1+x_2+x_3+x_4+x_5 = 20$. The total possible number of combinations is $\binom{19}4$. From there I subtracted the ...
1
vote
1answer
48 views

How to solve this combination?

I have $8$ pieces and $8$ places to put them into. I want to know how to calculate the number of possible combinations. The places are like this : $[a,b],[c(d,e),f(g,h)]$ $[a,b]$ is the same as ...
2
votes
1answer
21 views

Order of Binomial addition

I was reading up a statement in probability which said that $nC_0 + nC_1 + nC_2 +\dots+ nCm$ is of order $n^m$ where $nCm$ is notation for number of combinations from a collection of $n$ taking $m$ ...
-1
votes
1answer
19 views

Sum of fractions of coprime two integers [closed]

Let $a$ and $b$ be coprime two positive integers. For $a$ and $b$, consider two sets $A = \{ \frac{1}{a}, \frac{2}{a}, \dots, \frac{a-1}{a},\frac{a}{a} \}$ and $B =\{ \frac{1}{b}, \frac{2}{b}, \dots, ...
2
votes
0answers
41 views

Finding different sum factors of a number

Actual Question : A fair die is thrown k times. What is the probability of sum of k throws to be equal to a number n? My Work: Lets have k buckets, fill-in each bucket with value(1-6) so that the sum ...
2
votes
1answer
45 views

Where can I find a set of probability problems?

Is there a database of solved probability problems available? I am currently studying probability (and statistics) and, while I think I have a decent grasp of permutations, combinations, conditional ...
0
votes
1answer
15 views

Tag Rugby: 6 men a side, 4 on pitch, 2 subs. 40 minute game. How often to sub?

Playing a game of Tag Rugby tonight. We have 6 male players, of which 4 will always be on the pitch, with 2 subs. The game lasts 40 minutes, 20 each half. How long should each player get to play, ...
3
votes
6answers
128 views

How many $3$ digit numbers with digits $a$,$b$ and $c$ have $a=b+c$

My question is simple to state but (seemingly) hard to answer. How many $3$ digit numbers exist such that $1$ digit is the sum of the other $2$. I have no idea how to calculate this number, but I hope ...
0
votes
0answers
62 views

Choosing distinct elements from 5 sets

Given $5$ sets of each of them having $n_1, n_2, n_3 \dots n_5$ number of elements (all elements are integers from $1$ to $k$). What are the number of ways of choosing one element from each set such ...
3
votes
2answers
47 views

How many $3$ letter “words” consisting of at least $1$ vowel and $1$ consonant can be made from the letters of EQUATION?

The word EQUATION contains all five vowels. How many $3$ letter "words" consisting of at least $1$ vowel and $1$ consonant can be made from the letters of EQUATION? Hi, would anyone be able to ...
1
vote
3answers
27 views

Counting how many numbers have repeated digits

How many numbers are there in all from $6000$ to $6999$ (both $6000$ and $6999$ included) having at least one of their digits repeated ? (a) $216$ (b) $356$ (c) $496$ (d) $504$
1
vote
1answer
66 views

A 4 digits number is formed using 2,3,5,7 and 9 without repeat

A 4 digits number is formed using 2,3,5,7 and 9 without repeat. How many 4 digit numbers are there if each number has a remainder of 2 when divided either by 3 or 5? As i know, 2,3,5,7 and 9 is ...
2
votes
1answer
98 views

Number of ways possible to form a number?

Suppose we need to form a 4 digit number with the restriction that ...
-1
votes
0answers
43 views

No. of Combinations Possible

I have 'n'(<=10) arrays each containing some distinct numbers (from 1 to 100). I need to pick one no. from each array such that no 2 nos. are same. I need to find the total no. of such ...
0
votes
0answers
88 views

Selecting n different numbers from m sets of numbers?

we are given M sets of numbers.Each set has numbers ranging from 1 to 100.(there can be any number of elements in a set but all elements of one set are distinct (at max 100 elements)).we need to find ...
1
vote
0answers
58 views

Number of ways to answer three questions, with four choices each, and not get all of them right

I have this question, I could not get answer to it. In an examination there are three multiple choice questions and each question has $4$ choices. Number of sequences in which a student can fail ...
-1
votes
3answers
22 views

How many ways to fill six broadcast slots for 3 adertisements which are to be shown twice [on hold]

A television director is scheduling a certain sponsor’s commercials for an upcoming broadcast. There are six slots available for commercials. In how many ways may the director schedule the commercials ...
0
votes
1answer
42 views

Calculating combinations with colors

I need to know how many combinations there are possible from 10 balls with 5 colors. All colors must be in there, so in every combination there have to be 5 colors. Amount doesnt matter. Any tips? ...
0
votes
3answers
45 views

What is the minimum number of colours needed for coding 12 objects, if each may be marked with either one or two colours?

I have a word problem here which is a kind of high level to me A company that ships boxes to a total of (12) distribution centres uses colour coding to identify each centre. If either a single ...
6
votes
7answers
162 views

How many $10$ digit number exists that sum of their digits is equal to $15$?

How many $10$ digit number exists that sum of their digits is equal to $15$? Additional info: First digit from left is not $0$.we could use any digits from $0$ to $9$. I saw in some ...
2
votes
2answers
61 views

Number of ways to put one or more of $5$ books in $5$ bags

In how many ways can we put one or more of 5 books in to 5 bags? Additional info: books are labeled. Bags too. One or more bags can remain empty. Things I have done so far: There are $5$ ...
1
vote
0answers
33 views

Combinations of non-adjacent number sequences

I am trying to construct combinations of sequences. Given n integers (to choose from), say 1 <= n <= k and the requirements to construct sequences of length m. The number of sequences is simply ...
0
votes
2answers
33 views

Permutation and Combination with restrictions

There are 7 chairs. 4 are reserved for men and 2 are reserved for ladies. Remaining 1 can be occupied by anybody - either a lady or a man. In how many ways, can 6 men and 3 ladies be seated on ...
1
vote
3answers
32 views

Choosing $2$ groups of $5$ members and $1$ group of $2$ members from $15$ person

In how many ways can we choose $2$ groups of $5$ members and $1$ group of $2$ members from $15$ person? Additional info: groups are not labeled. Things I have done so far: I know number of ...
0
votes
2answers
31 views

Calculate the number of possible combinations

Given 4 sets of data, how can I find the number of possible combinations? Eg. Colors { Black, Blue, Red, Green } Shape { Round, Square, Triangular } Form { Solid, Liquid } Location { Inside, Outside ...
1
vote
1answer
77 views

Choosing 5 of 40 people sitting at a circular table so that between any two are at least three other people

$40$ people sit around a circular table. In how many ways can we choose $5$ people so that between any two of them there are at least $3$ other people? Things I have done so far: This question is ...
1
vote
2answers
74 views

Number of possible patterns?

Using the following rule: Each column and each row must contain at least one point, how many patterns can a 4x4 grid (thus with 16 possible point positions) generate? (this rule would thus make the ...
2
votes
1answer
52 views

What do you call a set whose subsets all have unique sums?

An example would be $\{1, 3, 7\}$, which has subsets with sums $1, 3, 7, 4, 10, 8, 11$. What is this called?
6
votes
3answers
140 views

Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
0
votes
1answer
29 views

To calculate number of combination of sequences having 1 and 2 alternating sequences of R and S.

I have a sequence of 6 letters containing 2 P, 2 R , 1 Q and 1 S. I have PPQ, now I have to add two R and one S in that, these can be placed anywhere. There will be total 60 possible ways to do that ...
0
votes
1answer
43 views

Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
0
votes
1answer
18 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
0
votes
1answer
47 views

Assigning $\pm 1$ values to the edges of a complete graph

I read this sentence in one combinatorics book. In graph $K_{100}$, there is a possible way to assigns number (value) from $\{+1,-1\}$ to each edge, so that the sum of all edge values connected to ...
-1
votes
3answers
43 views

Exactly why coefficient of $x^ky^{n-k}$ is $C(k,n)$ [duplicate]

in combination when we have a binomial lattices like $(x+y)^n$ the coefficient of $x^ky^{n-k}$ is equal with $C(k,n)$ ... for example we have $(x+y)^4$ so we have this $4$ factor ...
1
vote
1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
0
votes
1answer
65 views

Combination Problem with Sofa [closed]

Suppose we have 5 sofa on room A. in this room, 4 students seated on these sofa. These Strudents go to another room for eating dinner, and after that come back to room A. how many way the students can ...
0
votes
2answers
30 views

Ways to award first and second place to two persons out of nine

Question: In how many ways can the first and second place be awarded to two persons from among 9 people. ...
2
votes
0answers
47 views

What do you call the following operations on a symmetric matrix?

Suppose we have a symmetric matrix of the following form, where the diagonal is always zero: \begin{array}{cccc} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0\\ 0 ...
3
votes
1answer
72 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
3
votes
1answer
63 views

Chess Knight problem

Which is the number of all possible combinations of the knights, which are not mutually attack? The black knight may move to any of eight squares (black dots). The white knight in this case is ...
2
votes
2answers
34 views

Possible ways of 8 digit numbers using 1,2 and 3 such that the number has atleast one digit for each 1,2,and 3

How many 8 digit numbers can be formed using the digits 1,2 and 3 so that the number contains at least one of each of these three digits?
1
vote
1answer
39 views

Counting squares in a given k by k square..

So the question is : The solution to this problem according to the book is to first count the number of squares whose sides are parallel to the sides of this 10 by 10 square and then to count the ...
0
votes
1answer
62 views

what is the coefficient of following expression

what is the co-efficient of $x^{50}$ in the expansion of $$\frac{1}{(1-x^{1.7})(1-x^{1.8})(1-x^{2.6})(1-x^{3.0})(1-x^{4.0})(1-x^{6.7})(1-x^{7.5})(1-x^{8.2})}$$ can you please explain me the logic
0
votes
1answer
32 views

The number of ways of going up 7 steps …

The number of ways of going up 7 steps if we take one or two steps at a time is ? So its essentially asking in how many ways can we make use of numbers of (1,2) to get a sum of 7. Am I wrong up till ...