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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.

1
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2answers
29 views

Combinatrics/4 letter code word

Question reads: How many distinct $4$-letter code words can be made from the letters in the words "Pizza Pie" if the first letter must be a vowel and second must be a consonant. Answer is 98 I ...
4
votes
1answer
27 views

Counting the number of ways to form teams

Here's the Question, 12 employees are eligible for working on a company's project. The team must consist of 4 people. However, two employees had a fight and refuse to work together. How many ways can ...
0
votes
0answers
30 views

Adding the sum of 6 Dice, how do you calculate the permutations?

By working out the different combinations for the sum being 6,7 and 8, (1, 6, and 21) I can see this is a diagonal row in pascals triangle but I can’t work out how these would be calculated without ...
-3
votes
0answers
21 views

Reducing repetitions of combination [on hold]

Good day, i need some help with this problem: need count variants placements without repetition, for example, 4 objects(equal) if at least one on position 1-4, two 1-7, three 1-10, four 1-14. I can ...
2
votes
3answers
75 views

How many square matrices are there with given columns, rows and diagonals

Suppose we have $n\times n$ matrix that contains numbers from $1$ to $n^2$. How many matrices are there that their $n$ columns, $n$ rows and $2n$ diagonals contain given numbers? For example $3\times ...
1
vote
2answers
24 views

What is the probability that a set of nine children will contain three or fewer girls? [on hold]

I can't decide if it is 4/9 because there is the possibility of there being 0, 1, 2, or 3, or if it is 25% using combinations. Thanks!
3
votes
4answers
109 views

Selecting $2$ cards from a full deck of cards.

I thought of a problem earlier and I am quite clueless on how to solve it, or begin solving it, because I cannot find a way to easily compute the amount of combinations of $2$ cards that sum to a ...
2
votes
1answer
13 views

In how many ways can the sample be selected if it must have at least 2 male and 1 female mice?

A sample of 5 mice is to be chosen from 7 male and 6 female mice. In how many ways can the sample be selected if it must have at least 2 male and 1 female mice? I tried doing 7C2 * 6C1 but it seems ...
1
vote
2answers
39 views

What is the maximum number of subsets we can choose from a set of size 20 such that no two subsets have more than 2 common elemtns.

We have a set which has $20$ elements (eg. $\{1,2,3,...,20\}$). We want to choose the maximum number of non-empty subsets we can such that none of them has more than 2 common elements. for example $\{...
1
vote
1answer
35 views

What's the probability that picking $3$ coins from a bag containing all coins from $1$ cent through $2$ euro will yield a total of $80$ cents?

I recently ran into this interview question and am wondering if my solution is correct. The setting is We have a bag containing one coin of each type, i.e. we have one $1$ cent coin, one $2$ cent ...
2
votes
2answers
48 views

Variance and mean of balls in bins limited capacity

Let there be $m$ indistinguishable balls, $k$ bins, $C$ capacity. Let $X_j$ denote the total balls in bin $j$. I've seen ways to calculate the total number of combinations, but I'm not sure how to go ...
2
votes
2answers
61 views

Can you keep this raffle fair?

So there is a car draw in my area. There are 3221 participants in the draw. The winner is decided by a trustee drawing each digit from a separate drum. So from the first drum there is 0-3, the second ...
0
votes
1answer
47 views

Prove the identity $\binom{2n}{2}$ = $\binom{n}{2}+\binom{n}{n-2}+n^2$ where $n\geq2$ using a combinatorial proof.

Prove the identity $\binom{2n}{2}$ = $\binom{n}{2}+\binom{n}{n-2}+n^2$, where $n\geq2$, using a combinatorial proof. I've tried to think of it in terms of a counting problem. I think that for the ...
2
votes
2answers
93 views

Find the total number of 20 digit codes that can be formed using the numbers {0,1,2,3,4}, such that consecutive digits have a difference of 1?

To start with an example of such a code can be: $34321210123212343210$ I have no clue how this property can be mathematically counted. I actually even have a short solution of this question which I ...
0
votes
1answer
16 views

Number of combinations for a 4-character password with particular rules

We have a password with the following rules: 4 characters, no more, no less. Only normal alphabet characters (a...z) Only 1 uppercase character (but we don't know in which position). What steps ...
-2
votes
1answer
44 views

Combinations with Piano Keys

How many sound combinations can be created by the $10$ selected piano keys if each sound combination contains from $3$ to $10$ keys? Can anyone throw at least a hint? I am having difficulties with ...
-3
votes
0answers
29 views

Question Related to Probability of Two or more Events [closed]

A box contains $5$ red and $4$ blue balls. Two balls are down in succession with replacement from the box. What is the probability of getting: a. Red on the first draw. b. Red on the second ...
2
votes
2answers
43 views

can we perform modulo operator on a fraction on both of it's numerator and denominator?

I want to calculate nCr (mod $10^9+1)$.so for calculating nCr we have: $$nCr=\frac{n!}{r!(n-r)!}$$ so I want to know whether it is true that I perform modulo operator to numerator and denominator ...
1
vote
1answer
38 views

Weird lottery proabablity question

In this lottery $7$ balls are chosen from $1-60$. In order to win the main prize, you must select all $7$ right. I calculate the odds of doing this as: $$1:386,206,920$$ The odds of getting $3$ ...
0
votes
1answer
19 views

Finding optimal subset of items with a blackbox reward function

Given a set of $N$ items, I would like to find an optimal subset of a maximum number of $k$ items that maximizes a blackbox reward function $f(subset)$. I want to find a global maxima efficiently by ...
-2
votes
1answer
44 views

How many quadrilaterals are there? [closed]

I use about nCr but stuck at when 3 points when i chosen is on the same line. Help me please.
1
vote
2answers
42 views

In a word containing $k$ A's, how many permutations place at least $n$ A's consecutively?

Suppose a word is $l$ letters long, and it contains $k$ A's. (The specific letter is irrelevant) Is there a general formula to count how many permutations contain at least $n$ consecutive A's? (Assume ...
-2
votes
1answer
15 views

How many different arrangements are possible? Combination & Permutations [closed]

Janet has 10 different books that she is going to put on her bookshelf. Of these, 4 are Chemistry books, 3 are Biology books, 2 are Statistics books, and 1 Physics book. Janet wants to arrange her ...
1
vote
1answer
28 views

Paths on a grid and how many [duplicate]

You start at coordinate (0, 0) on a grid and want to reach position (3, 3). On this grid, you can only move right or up, not diagonally. How many paths are there? This is a question I am trying to ...
0
votes
1answer
25 views

Is there a term for permutations where the elements are optionally included?

The permutation for abc would be: abc acb bac bca cab cba But if the elements are optional: ...
0
votes
1answer
20 views

Finding the number of combinations under group restrictions

Suppose I have 4 instruments, which ranges from 0-100 with steps of 25. For example, Instrument 1 can have 5 scenarios, 0, 25, 50, 75, 100 Similarly, instruments 2-4 can also have same scenarios. Now,...
1
vote
1answer
21 views

Given $N$ slots and $S$ objects to fill those slots, how many ways are there to fill the slots such that no two objects are adjacent.

Given $N$ slots and $S$ objects to fill those slots, how many ways are there to fill the slots such that no two objects are adjacent? I can't see a general pattern for this. If I take $N=7$ and $S = ...
0
votes
2answers
26 views

Find the number of ways to select two balls of different color through combinations

In a box there are 3 green, 5 red, 7 yellow and 6 blue balls. Find the number of ways to select 2 balls of different colours. The answer is 161. I am clueless on how to go about this question.
0
votes
0answers
26 views

probability of dice rolls, the way to write set of outcomes

So, assume that we roll a die until 3 turns up. Let $A$ be the event that the first time a 3 turns up is after an even number of rolls which means odd throw. How do we find and write the set of ...
0
votes
1answer
15 views

Combination question, number of ways 5 men and 5 women can marry

What is the number of ways $5$ men and $5$ women can marry? (if each person can only marry one person from opposite gender) I have seen $5!$ because each gender can start from a person being able to ...
1
vote
2answers
28 views

Baseball Probability With Permutation and Combinations.

I am a (Gifted) sixth grader, who after taking the government issued MAP test, got selected test questions based on my weaknesses. One of the test concepts I got was probability, but it was Geometry ...
-1
votes
3answers
87 views

Choosing 2 squares on an $8 \times 8$ ($64$ square) chessboard

On an $8 \times 8$ ($64$ square) chessboard, how many ways can we choose pairs of squares such that each pair doesn't have the same colors? Each pair should consist of a white and black square.
10
votes
2answers
581 views

ways of selecting consecutive persons sitting at a table

I'm trying to solve the following problem: Ten people are sitting around a round table. Three of them are chosen at random to give a presentation. What is the probability that the three chosen ...
0
votes
0answers
23 views

Calculating combinations with Duplicates and no Rearangements [duplicate]

I am wondering how to calculate the number of combinations of $n$ amount of decimal numbers, and $p$ represents amount of numbers chosen out of $n$, where rearrangements are not allowed but duplicates ...
-1
votes
0answers
43 views

Arrangment Problem

Compute the number of different ways you can tile a 10x32 (height=10, width=32) rectangle using 1x2 and 1x3 tiles and complying to the following constraints: The rectangle should consist of 10 rows ...
1
vote
0answers
17 views

Find the number of ways to choose $k$ objects from a set of $n$ objects arranged in a circular order such that no two consecutive elements are chosen

Find the number of ways to choose $k$ objects from a set of $n$ objects arranged in a circular order such that no two consecutive elements are chosen. I could think of a combinatorial solution- ...
0
votes
1answer
18 views

Probability that the blocks have 2 colored faces each

A cube with all six faces colored is cut into 64 cubical blocks of the same size which are thoroughly mixed. Find the probability that the 2 randomly chosen blocks have 2 colored faces each? With ...
3
votes
2answers
31 views

Combinations formula shorthand with $2$ as the base.

Hope you are doing great. I hope you will help me to understand one thing related to combinations. I have grasped general ideas behind permutation and combination, but I can not understand how the ...
2
votes
1answer
36 views

Combinations & Probability - Sport Club Teams Probability - Is my solution correct?

I'm revising permutations, combinations and probability for an upcoming exam and would really appreciate if someone could take a look at my procedure to solve this problem and let me know if it's ...
0
votes
0answers
21 views

Combinations for data set with repeated elements

For a data set of integers from 1 to 48, $$\binom{48}{4} = 194,580$$ distinct combinations. And for another data set of 48 integers with 4 subsets of integers from 1 to 12 in each subset, a $$\binom{...
-1
votes
1answer
75 views

Finding the sum of a series that is a combination [closed]

I have to find the sum of this series $$\sum_{k=0}^n \binom{2n}{k}$$
0
votes
2answers
43 views

2D grid of size 3*n

There is a 2D array of size 3*n. Suppose there are 3 numbers 1, 2 and 3. What can be the number of ways in which we can put numbers in 2D array using these numbers only according to below rule 1)...
1
vote
3answers
62 views

How many ways can 26 letters be divided into 13 pairs?

The question is: a plug board has 26 letters, and there are 13 cables. The cables connect all possible pair of letters. How many possible configurations does the plug board provide? In other words, ...
0
votes
2answers
24 views

What is best covering(s) of all 190 pairs in [1..20] range by minimal N 4-number combinations and how can they be generated/constructed?

We make combinations of 4 distinct numbers (without repetition) out of $[1..20]$ integer range. We need to identify minimal (in terms of number of 4-combinations) combination sets which cover all $...
0
votes
2answers
46 views

Simplification of a sum involving binomial coefficient

Let the following complicated sum: $$\sum_{n=1}^k \frac{1}{2^{2n+1}}\binom{2n}{n}.$$ Is there a way to simplify it? Or should we upper and lower bound it? Thank you
0
votes
1answer
43 views

roll 4 dice and disgard the lowest, what's the probability of the sum being over 15

I'm wondering if there is a neat way to do or set up a simulation to check my answer. I tried brute force, looking at the number of 666a rolls (21) then the number 665a rolls etc. using multinomial ...
0
votes
0answers
19 views

Calculating the proportion of combinations by number of unique objects within

Essentially I would like to figure out a formula for determining the proportion of combinations in a series with 'n' objects and 'r' samples with replacement (i.e. CR(n,r)) by the number of unique ...
3
votes
2answers
287 views

What is the relation in this sequence

I was attending a coding competition and this question came up There is a box containing n chocolates. You can either take 1 chocolate or 3 chocolates at a time until its empty. So in total how many ...
1
vote
2answers
22 views

Binary Combinations Less Repetition

What would the formula be for finding the number of combinations of $n$ binary elements when no $0$ can follow another but there is no restriction on subsequent $1$s. For example, an allowable ...
3
votes
2answers
37 views

Combinatorics - Limited Replacement

A man is selling $15$ different items, and has exactly $3$ of each item. They all cost the same and we have enough to buy $7$ items in total. How many combinations of items can we buy? I know that if ...