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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Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

Example: Uniquely distribute 2 Red Balls and 4 Blue Balls into 3 boxes: [B][BB][RRB] [B][BBB][RR] [B][R][RBBB] [B][RB][RBB] [BB][R][RBB] [BBB][R][RB] Answer: ...
1
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1answer
39 views

Cartesian product with all elements

I have two sets A and B with $A = \{1,2,3\} \\ B = \{ A, B, C, D, E \}$ Now I need to get something similar to the Cartesian product. If my understanding is correct, the Cartesian product would ...
0
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1answer
24 views

Mixing up seating charts: Measuring “mixedness” over time

Background: My class has $10$ students and $3$ tables; naturally, the students are distributed with $3, 3,$ and $4$ seated at the individual tables. On the second day of class, students sat in the ...
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0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
0
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4answers
43 views

All possible combinations

I have two sets (1,2,3) and (A,B,C,D,E). I want to calculate all possible combinations. This would be my approach: combinations with a single 1: ...
1
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0answers
32 views

Combination estimation

I am looking at a proof where part of it derives an estimation for the number of combinations and I cant understand how the following step is derived: $\varepsilon^{-\varepsilon L}\underset{j \leq ...
0
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1answer
26 views

Combination during time period

There is a workforce who can handle $3$ products and the $3$ products have different execution times: $1h$, $2h$, $4h$. How do I calculate all possible combinations this workforce may create between ...
4
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3answers
335 views

Students in a class, girls sitting with boys and boys sitting with girls

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
0
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1answer
57 views

Number of combinations where the sum of values must be the same

My question is as follows: let there be $n$ different numbers $a_i$ in a set $A$, where each $a_i$ is a number between 0 and 1. How many different sets of values can I have that fulfill the condition ...
0
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3answers
55 views

Counting numbers of possible solutions

For the equation $\displaystyle x_1+x_2+x_3+x_4+x_5=n$ there are $\displaystyle \binom{4+n}{4}$ solutions. But what about the equation $\displaystyle x_1x_2x_3x_4x_5=n$ ? Assuming $\displaystyle ...
2
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1answer
61 views

How to find out the probability of an event about which we have two informations

I would like to know how to find out the probability of an event about which we have two informations. Say we have $A$ and we know it is lower than $K$ but greater than $X$. How do you find the result ...
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1answer
13 views

Combination question, choosing from subgroups without replacement

I have 8 groups, each each group has 3 distinct elements, must select 1 element from each group. How many unique arrangements can I make, selecting 1 element (of 3 choices) from 8 groups? Is it ...
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0answers
26 views

How do I calculate all possible combinations for a player creator in a game?

I'm currently working on a character creator for a game, but I don't know how to calculate all possible character combinations the player can create. In the creator, the player is required to choose ...
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2answers
33 views

Arrangement of all the letters of a word [on hold]

In how many ways can all the letters of the word ‘PERFORMED’ be placed in the $3 \times 3$ grid of squares, such that each square contains exactly one letter and there is at least one vowel in each ...
0
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1answer
27 views

Balls and Boxes Generalization

Recently, I saw a problem here on MSE: $$$$"Put 9 pigs in 4 pens such that there are an odd number of pigs in each pen." Individual cases or solutions to the problem are quite easy. But how would we ...
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0answers
27 views

Suppose a sequence of 8 nucleotides contains 2 each of A, C, G, T. How many such sequences are there? [closed]

Suppose a sequence of 8 nucleotides contains 2 each of A, C, G, T. How many such sequences are there? please post the solution..
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2answers
45 views

How many ways can the players enter? [closed]

$5$ players want to enter a stadium through three gates of the stadium, However, each gate of the stadium can only pass two players. How many ways can the players enter the stadium?
6
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1answer
62 views

Poker Combinations: How many ways can you get 4 of the same suit in a hand of 5 cards?

The homework question is: in how many ways can we get exactly 4 cards of the same suit in a hand of 5 cards? (Order does not matter.) Here is what I have: we need to pick two different suits, decide ...
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1answer
46 views

What is the combinatorial proof for the formula of S(n,k) - Stirling numbers of the second kind?

What is the combinatorial proof for the formula of Stirling numbers of the second kind ? i.e. S(n,k) where n is the number of objects and k is the number of parts $${n\brace ...
2
votes
1answer
30 views

How many ways to put 8 rooks on the chessboard which satisfy…

How many ways to put 8 rooks on the chessboard which satisfy that there is no rook can attack the others rook, and there is no rook place in : 1) One main diagonal. 2) Bot Two main diagonal My ...
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0answers
11 views

Boundedness of binomial coefficient multi-index [hold-on]

Let be $\alpha,\beta\in \Bbb{N}^n$ multi-index, such that $|\alpha|=m+1$ e $\beta <\alpha$, with $m\in\Bbb{N}$ fixed. Is it possible to bound $\large\large{\alpha \choose \beta}$ by something that ...
3
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2answers
55 views

In how many ways can you group $3$ different numbers from $1$ to $12$ wherein their sum is divisible by $3$?

In how many ways can you group $3$ different numbers from $1$ to $12$ wherein their sum is divisible by $3$? This question is one of the questions asked in a Math contest for intermediate level, ...
5
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2answers
36 views

How to rearrange these numbers?

Say you had to add up to 10 by only used the positive integers of 1 and 2, they can be arranged by 1111111111 as the most digits needed and 22222 as the least, or say 2212111 would be another ...
7
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1answer
100 views

How many ways to add to 32?

I have been presented with a rather complex combination problem. Using only the numbers 2, 4, 6 and 8, how many possible ways can you add up to 32 if the number 4 may only be used no more than once ...
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1answer
19 views

Number of ways of selecting all k-indexed identical items before all k+1 indexed identical items for all k from 1 to n

Suppose we have n indices and we have a specific number of items allotted to this index. Say for 2 balls of colors Blue(B)[1], 4 of color Green(G)[2] and 2 of color Red[3] (I could've just assigned ...
0
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1answer
57 views

Probability of (A <= B OR A <= C) AND B > C when A, B and C are random integers with different ranges but starting at 0.

I have 3 random integers A, B and C, along with 3 defined integers X, Y and Z: A in [0, X] B in [0, Y] C in [0, Z] All the values that A can take within its defined range are equiprobable. Same goes ...
0
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1answer
42 views

Probability of A > B AND A > C when A, B and C are random integers with different ranges but starting at 0.

I have 3 random integers A, B and C, along with 3 defined integers X, Y and Z: A in [0, X] B in [0, Y] C in [0, Z] All the values that A can take within its defined range are equiprobable. Same goes ...
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2answers
43 views

Should I be using combinations or permutations?

I have a set of $26,000$ values. Each value has the option of being $1$ or $0$. How do I calculate the number of potential combinations of $1$'s and $0$'s that exist for $26,000$ values?
4
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0answers
64 views

About two combinatorial counting problems.

Here are the problems: Suppose $X$ is a set of $n$ elements, and $S_1,...,S_m$ are $m$ subsets of $X$ of average size at least $n/w$. Show that if $m\geq 2kw^k$, then there are $k$ distinct ...
2
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1answer
50 views

In how many ways 3 numbers can be chosen from a from the set {1, …, 18} so that their sum is divisible by 3?

In how many ways 3 numbers can be chosen from a from the set {1, ..., 18} so that their sum is divisible by 3? Now, I've seen the solution, but I can't get my head around one detail. The solution ...
0
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2answers
52 views

Applying Inclusion-Exclusion principle

How to apply principle of inclusion-exclusion to this problem? Eight people enter an elevator at the first floor. The elevator discharges passengers on each successive floor until it empties on ...
0
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2answers
28 views

How many combinations can you get from a three times three matrix

I have a 3*3 matrix like this (figure 1): * * * * * * * * * Slots can be filled similar to next examples (figure 2): ...
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0answers
37 views

Can we find the closed-form of the series?

I want to calculate the series $$ F(N,g)=\frac{1}{g^N}\sum_{m=0}^{N(g-1)}\Big(\sum_{i=0}^{[m/g]}(-1)^i\binom{N}{i}\binom{N-1+m-gi}{N-1}\Big)^2 $$ where $g=2,3,4,\cdots$, and $N$ is any positive ...
0
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1answer
20 views

Special Form of Combination - Formula Verification

Giving abc how many combination that involves a ? answer is a ab ac abc equal to 4 I came up with the following formula but I would like to know of its correctness plus if there is simpler form ...
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2answers
25 views

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least one ball?

In how many ways can $5$ identical balls be placed in the cells of a $3 \times 3$ grid such that each row contains at least 1 ball? I proceeded like this- In the first row choosing one cell out of ...
4
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1answer
24 views

Prove that for any prime $p$, if $a>b$ then $p^2$ divides $C(pa,pb)-C(a,b)$.

Let, $p$ be a prime and $a>b$. If $\operatorname{C}(n,r)$ denotes the combination of $r$ objects from a collection of $n$ objects taken at a time, prove that ...
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2answers
21 views

Permutation with constrained repetititons

The question is as follows: How many ways can 12 identical white and 12 identical black pawns be placed on the black squares of an 8 x 8 chessboard My answer was $\frac{32!}{12!*12!}$ But the ...
0
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1answer
43 views

$\frac{(-1)^n}{2\cdot 4\cdot \cdot\cdot2n}=\frac{(-1)^n}{2^n\cdot n!}$

$$\frac{(-1)^n}{2\cdot 4\cdot \cdot\cdot2n}=\frac{(-1)^n}{2^n\cdot n!}$$ $$\frac{(-1)^n}{3\cdot 5\cdot \cdot \cdot(2n+1)}=\frac{{(-2)^n} \cdot n! }{(2n+1)!}$$ can anyone tell me if these are true or ...
0
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1answer
11 views

How to find all possible groups of four different values(integers)

I have four values :50,100,500,1000. I want to know many groups could be made with this combinations values. 50,100,500,1000 here it would be count as 1+1+1+1 50,50,50,100 count= 3+1 ...
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2answers
30 views

Marbles Combinations problem

Martin’s bag of marbles contains two red, three blue and five green marbles. If he reaches in to pick some without looking, how many different selections might he make? I do not know how to ...
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0answers
25 views

In all normal magic squares of size 4 by 4 what is the distribution of values 1,2,3…16 in all positions irespectable of rotations and reflections

when i say irrespectable of rotations reflections i mean that all corners are essentially the same position and so on... c a a c a b b a a b b a c a a c i want to know for all 880 distinct ...
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1answer
28 views

Terminology for splittings of a set into two parts

I have a set of values $V$ that can be split by any combination $C$ of the elements $v$ that belongs to $V$. Order is not important and repetitions are not allowed. For example, $V := \{1,2,3,4\}$ ...
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4answers
28 views

Unique Combinations

I hope someone can help me with some combinations (and perhaps permutations). This is still the hardest area of math for me, but I'm still trying. This is a two part question. (1) I have a bag of ...
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2answers
29 views

What is the probability that a boy who knows how to solve $25$ of potential $30$ questions will get at least $8$ of $10$ correct?

A boy is preparing for test. The teacher gives $30$ questions to study from and will select $10$ out of the $30$. The student only know hows to solve $25$ of the $30$ questions. A)What is the ...
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2answers
38 views

Permutation of 6-digit numbers without repetition

How many 6-digit numbers without repetition of digits are there such that a ) the digits are all non-zero b ) 1 and 2 do not appear consecutively in either order ? Calculated the answer as below ...
3
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1answer
44 views

There are 39 students in our class. We form groups of 2, with one left out. How many ways can the students be paired up?

I know that the answer is $\frac{39!}{(2!)^{19}\cdot19!}$, where each pair can be organized $2!$ ways and the pairs can be arranged in $19!$ ways. We can also extrapolate the case for $5$ students, ...
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2answers
41 views

An elevator starts with 10 people, how many ways can all the people… Cases for each floor?

An elevator starts with 10 people on the first floor of an 8 story building and stops at each floor. In how many ways can all the people get off the elevator? The only way I can think to do this is ...
3
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3answers
245 views

How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number?

How many four-digit odd numbers, all of digits different, can be formed from the digits 0 to 9, if there must be a 5 in the number? I know that there are 4 different cases where 5 is in the ...
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1answer
58 views

Obtain all combinations of 3 numbers with repetition.

I'm stuck with this problem and I'd like to get some help. I think there is something I'm not aware of. So, the thing is I'm given this control matrix H. ...
0
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1answer
18 views

Choosing $8$ shoes from $20$ different pairs of shoes to get at least two pairs

Let's say I have $20$ different pairs of shoes in my wardrobe. I want to know, in how many ways can I choose $8$ shoes, so that I have at least two pairs? All I know is that I can choose $8$ shoes ...