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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.

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Odds of the perfect game of bingo.

Playing breaking bingo. The last round is a jackpot round where the caller calls seven balls containing at least a B, I, G, and O. (The freespace provides the N if no N is called). To win the Jackpot, ...
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1answer
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Expected number of rounds, adding $g$ rounds with probability $r$

This is closely related to the one on the link below, but I cannot really map it to mine... I play a game, with probability $r$ the game adds $g$ more rounds. What is the expected number of rounds ...
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3answers
64 views

The number of integral solution of $\alpha+\beta+\gamma+\delta$=18 such that..

Question The number of integral solution of the equation $$\alpha+\beta+\gamma+\delta=18$$, with the conditions: $1\leq\alpha\leq5$; ${-2}\leq\beta\leq4$; $0\leq\gamma\leq5$ and $3\leq\delta\...
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0answers
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Expected value: random prizes until occurrence of termination 'prize'

I have 10 boxes of which 9 have a coin and 1 is empty. I can choose boxes and keep the prizes until the empty one appears. What's the expected value? First I thought: $E$ must the the mean between ...
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0answers
14 views

construction bijection and to understand Dyck path.

In here: https://oeis.org/A080936. I want to understand $T(n,k)$ is the number of Dyck paths of semilength $n$ and height $k$ and following triangle. \begin{align} 1;\\ 1, & 1;\\ 1, & 3, 1;\\ ...
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1answer
42 views

Sum of all possible numbers? One has to only for solely $5$ digit numbers.(for example $05793$ does not count) [on hold]

What is the sum of all possible numbers formed using the digits ${0,3,5,7,9}$ without repetition? I’m particularly confused when they ask the sum with one of the digits as $0$.
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1answer
27 views

Combinatorics of two pairs in poker

I've read the answer on counting the number of possible two pair hands in poker and I understand it. However, I don't know why my initial approach was wrong. Can someone point to the flaw in my ...
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1answer
42 views

probability - selecting balls from a box

A box contains 4 white, 5 red, and 6 black balls, four balls are drawn at random from the box. Then the probability that among the balls drawn there is at least 1 ball of each color is? What goes ...
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1answer
19 views

Permutation and combination Number of Bits Problem [on hold]

how many bit strings of length eight either start with a '1' bit or end with the two bits '00'? I got the answer 128. Is it correct and If not please do explain??
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1answer
35 views

Find number of triangles with integral sides and side lengths ≤ 2n

Find number of triangles with integral sides and side lengths less than or equal to $2n$. I approached this method by recursion. Say $A_{2n}\ $is the number of triangles with integral sides and side ...
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2answers
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Combinatorics - How many ways to partition an integer n into k bins with values 0 - 5 and restrictions

I have been struggling with this brain teaser for some time now. I looked at some combinatorics and partition equations but I can't find the one that captures the solution entirely. Frame I have a ...
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3answers
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Why my process is wrong:-How many ways are there to choose $5$ questions from three sets of $4$, with at least one from each set?

Question A question paper on mathematics contains $12$ questions divided into three parts A, B and C, each containing $4$ questions. In how many ways can an examinee answer $5$ questions, selecting ...
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1answer
50 views

Combinatorial proof $\sum_{i=k}^n {i-1 \choose k-1} = {n \choose k}$ [duplicate]

Could someone help me as I am stuck with coming up with a proof for this? Assume n is the total number of people in a town. Assume k is the number of possible ways to select a chief of the town. So ...
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2answers
25 views

Isn't Combination just selection?

So, There is this question which I came across. "Total no of handshakes among 15 people." The answer seems to be just 15C2. Isn't that just a way of selecting and not the number of handshakes? I'm in ...
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3answers
41 views

Consider an urn that contains $4$ blue and $7$ red balls

I'm in need of some help with this problem. Consider an urn that contains $4$ blue and $7$ red balls. First one ball is selected, and then a second ball is selected without replacement. (a) What is ...
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1answer
40 views

Combinations without repetition and with limit on consecutives

I see from Wikipedia that Binomial coefficient finds the number of k-combinations from a given set of n elements. In this figure ...
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0answers
37 views

In how many ways can the letters of the word ALGEBRA be rearranged?

In how many ways, can the letters of the word ALGEBRA be rearranged with condition: the two As never appear next to each other. The way I think of a solution is: total number of permutations of the ...
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1answer
34 views

Combinatorial problem on every day expense [on hold]

We have $n$ dollars. Every day we buy exactly one of the following products: pretzel ($1$ dollars) candy ($2$ dollars) ice-cream ($2$ dollars) soda ($2$ dollars) bubble gum ($2$ dollars) cracker ($...
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2answers
22 views

No Of ways of arranging plates belonging to two different sets?

There are $n$ triangular plates on which numbers from $1$ to $n$ are written and there are $m$ circular plates on each of them number from $1$ to $m$ are written. We have to find the total no ways of ...
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2answers
32 views

How many 4 digit numbers can be formed by digits 3, 1, 3, 1? [on hold]

This a question from my textbook and it says the answer is 6 and I fail to see how. Per my understanding if we consider each digit being used only once i.e viewing all 4 digits as distinct elements (...
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2answers
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Probability of successful Secret Santa selection

In the following Secret Santa Scenario: There are 12 people, each have written their name on a scrap of paper and put it into a bowl. The bowl is then passed around and each of the 12 picks out a ...
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1answer
90 views

What does this symbol denote? [duplicate]

I saw this symbol: $\underline{\big|6} $ in a question bank for a chapter of permutations and combinations. I have included the question from the book to provide more context: image link.
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1answer
37 views

Let $A_p = \{ d\in \{1,2,\dots, 999\} : p^{m} \| d \text{ where $m$ is odd} \}$. What is $|A_p|$? [on hold]

For any prime number $p$, let $A_p$ be the set of integers $d\in \{1,2,\dots, 999\}$ such that the power of $p$ in the prime factorization of $d$ is odd. Then which of the following options are ...
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1answer
40 views

Eight fair coins are flipped. Find the probability that 5 heads and 3 tails are obtained

I am struggling to solve this probability question. I don't know which approach to take with regards to finding the equation for calculating the given amount. $${8\choose5}\left(\frac12\right)^1\left(...
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2answers
41 views

If $12$ distinct points are placed on a circle and all the chords connecting these points are drawn, at how many points do the chords intersect?

If $12$ distinct points are placed on the circumference of a circle and all the chords connecting these points are drawn, at how many points do the chords intersect? Assume that no three chords ...
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2answers
23 views

Question on word combinations with exclusivity

"How many 4 letter words on the alphabet {a,b,c} in which 'a' occurs exactly twice are there?" My answer is incorrect as I answered 3*3*2*2 4 letter words. However, this doesn't necessarily remove '...
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0answers
29 views

Counting with repetition, how to determine which one is r and which one is n

* I know and understand that there are $\binom{n+r-1}{r-1}$ or $\binom{n+r-1}{n}$ ways to distribute n identical candies to r children. But sometimes, I can't determine which one is r and which one is ...
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0answers
29 views

What is the probability of getting not more than 500 utopians?

The Utopia population: 602427 inhabitants of which 313839 are of utopian nationality. We choose 1000 inhabitants. What is the probability of getting 1) not more than 500 Utopians 2) at least 400 ...
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1answer
57 views

Finding sum of digits of $m$ [closed]

If the sequence of 5 positive integers (a,b,c,d,e) satisfy: $$abcde\leq {a+b+c+d+e} \leq 10m$$ then find the sum of digits of m. I don't know how to approach this question. I know it's not a good ...
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1answer
54 views

Why can’t we use multinomial theorem here?

We have $10$ white, $9$ green and $7$ black balls. All balls are identical except for colour. While the solution for selecting number of ways in which one or more balls can be selected from these ...
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1answer
40 views

Binomial distribution: what is the probability of getting exactly 3 women from a draw of Y = 1 to 10?

So my question is We choose a certain number Y of different people what is the probability of getting exactly 3 women from a draw of Y = 1 to 10 ? what is the probability of getting at least more ...
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0answers
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How many combinations of team I can make with limited credit limit? [closed]

How many combinations if team I can make if I have to pick 11 players within 100 credits, each players are assigned certain credits, for example Player A has 10 credits, Player B has 8.5 credits etc. ...
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1answer
29 views

If there are four bakeries that each close one day a week, how many schedules are possible if at least one bakery is open each day?

I’m self studying for a probabilities and statistic exam so unfortunately, i don’t have anyone to ask. So the question goes, we have 4 bakeries and 7 days in a week. Each bakery closes once a week. a)...
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1answer
27 views

How many ways are there? [closed]

How many ways are there to put 14 identical objects in 3 distinct boxes with at least 8 objects in one box? What is the thinking procedure of the similar question type?
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1answer
28 views

How many ways can a number be written as a sum of two non negative integers?

How many ways can a number be written as a sum of two non negative integers? For example there is $4$ way for $7$. $ 7=0+7=1+6=2+5=3+4$ I think there is $[ \frac{N}{2}]+1$ way for number$N$. Is ...
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Combinatorial Optimization: techniques to reduce variance of a series of angle values

I am working with a series of angle value $\Theta = [\theta_{0} ~ \theta_{1} \cdots \theta_{N-1}]$. Each $\theta_{n}$ has a value belonging in $[-\pi;\pi]$. My problem is that I would like to reduce ...
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4answers
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Review on my method for $Number$ $of$ $diagonals$ in a regular $n$-gon is $\frac12n(n-3)$

I have an assignment on permutations and combinations topics. In that there is a question- The number of interior angles of a regular polygon is $150^\circ$ each. The number of diagonals of the ...
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0answers
78 views

Find a good formulae

We know that equation $$s_1+s_2+s_3=n-1 \quad \mbox{$s_1,s_2,s_3$}\geq 1$$ has $\binom{n-2}{2}$ solution. I want to find any good formulae for the following form : $$\sum\limits_{(s_1,s_2,s_3)}\...
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3answers
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When do i use the power ^ sign in a combination question?

My problem: If there are 5 different candies in a jar and a child wants to take out one or more candies, how many ways can this be done? I said it is $^5C_1 -\; ^5C_0 = 5-1 = 4$ ways. The $-1$ for ...
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1answer
27 views

Permutations & Combinations: Probability & Statistics [closed]

thank you for stopping by and thanks in advance for your time! I'm struggling with a probability & statistics problem while studying for my exam so any help would be much appreciated and even more ...
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1answer
18 views

How many 5-card hands from a standard 52-card deck contain exactly 1 king and exactly 1 heart?

Is my solution correct? There are two cases: Case 1: (the chosen heart is not a king) So there are $3 \choose 1 $ ways to choose the king, and $12 \choose 1 $ ways to choose the heart, and then $...
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2answers
21 views

Discrete math counting principles

In how many ways can a teacher distribute 12 identical science books among 15 students if 1) no student gets more than one book? 2) if the oldest student gets two books but no other student ...
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1answer
29 views

Permutations vs Combinations: Probability & Statistics

I need help with the following exercise as I am preparing for my exams on my own and never had the chance to visit any workshops or lectures as they were not offered, so here it goes. A lock can be ...
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1answer
23 views

How many intersecting there's between Diagonals and smaller parts?

If there is a rectangle to the side $a$ and $b$. $(a \leq b)$. Then divide it into $ab$ smaller segments, and then draw the rectangle diameter. How many intersections are there between Rectangular ...
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3answers
31 views

All possible ways to split a number

I am trying to find a way to find (if it is possible) how many ways there are to split a number of n digits considering that the "splits" can occur everywhere and the subsets don't have to be the same ...
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1answer
22 views

Selecting council members for a committee - elementary combinatorics.

I'm trying to figure out how to do the following question, but I got stuck. I just don't see how they are counting these people. In a student council consisting of 16 persons there are mathematics- ...
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1answer
26 views

How many random 3 letter words have exactly 1 “A”? No repetitions. Trying to build intuition here

I think the options are either: c(3,1) * c(25,2) (as in, choose 1 of the three letters to be the “A” and choose 2 of the remaining 25 letters for the other one) = 900 OR if the first letter is an A,...
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2answers
23 views

Ways of selecting $3$ balls out of $9$ balls if at least one black ball is to be selected

A box contains two white, three black, and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw. The answer given is $64$. I ...
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How many strings of four decimal digits do not contain the same digit twice

The answer should be 1: Four different digits : $$10*9*8*7 = 5040 $$ 2: Four different digits + contains the same digit three times + all digits are of the same : $$ 10*9*8*7+10*{4 \choose 3} *9 + ...
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3answers
51 views

Four men go into a restaurant and leave their umbrellas at the door

Question: Four men go into a restaurant and leave their umbrellas at the door. On their way out, each man picks up an umbrella and they discover when they get outside that no man has his own ...