0
votes
0answers
26 views

How many Possible Combinations exist?

I have $120$ coins and $21$ buckets. Each bucket can hold $0$ to $20$ coins. How many possible coin/bucket combinations are there?
0
votes
2answers
22 views

Proving $ \sum_{r=0}^{n} \frac{1}{r+1} \binom{n}{r} = \frac{1}{n+1} (2^{n+1} - 1) $

I'm stuck at proving the following. $$ \sum_{r=0}^{n} \frac{1}{r+1} \binom{n}{r} = \frac{1}{n+1} (2^{n+1} - 1) $$ This is what I have so far. $ \sum_{r=0}^{n} \frac{1}{r+1} \binom{n}{r} = (1) ...
0
votes
1answer
23 views

Problem finding the number of r-element multi-subsets of the multi-set $M=\{ a_{1},a_{2},…,a_{n},m.b \} $

Let $m,n,r \in \mathbb{N}$. Find the number of $r$-element multi-subsets of the multi-set $$M= \{ a_{1},a_{2},...,a_{n},m.b \} $$ when $r \leq m,r\leq n$. Below is the given answer. ...
1
vote
2answers
30 views

Probability of being selected twice in a week given a set of n people?

Let's say a child is selected out of a group of 10 students each day to stay after school and help clean the classroom. What is the probability that a particular child is selected exactly twice during ...
1
vote
3answers
63 views

Problem proving $ P_{r}^{r} + P_{r}^{r+1} + … + P_{r}^{2r} = P_{r}^{2r+1} $

Show that $$ P_{r}^{r} + P_{r}^{r+1} + ... + P_{r}^{2r} = P_{r}^{2r+1} $$ where r is a nonnegative integer. This is what I've come up with so far but I'm not sure how to continue. I know I need to ...
0
votes
0answers
15 views

Combinatorics or permutations of language? [on hold]

I am looking to understand the combinatorial or permutative space of the English language. 26 letters up to length 25. How big is the space? And if anyone has an algorithm for this in shell or ...
0
votes
0answers
34 views

Distributing balls in boxes.

In how many ways can $n$ identical balls be distributed amongst $m$ different boxes given that a box can have any number of balls(from $0$ to $n$)? What I've tried is using multinomial theorem to ...
0
votes
3answers
53 views

Need help with flaws in statistical reasoning

The problem is as follows - there are three couples and six chairs in a row. The six individuals are seated at random. What is the chance that at least one couple will be seated together? Here's my ...
2
votes
2answers
102 views

Hat Matching Problem Expectation

I have an interesting problem in the context of the hat matching problem: There are n people with hats at a party. Each person randomly grabs a hat. A match occurs if a person gets his own hat. I'd ...
0
votes
1answer
15 views

Find the possible number of assignments?

S students, I interviewers, each student has to undergo R interviews, each interviewer can interview at max X students. No student interviews with an interviewer more than once, and no interviewer ...
0
votes
1answer
32 views

A Question on distribution numbers

This is a question from the book Combinatorics -a problem oriented approach which states: Q.1 Find the no. of distributions of a set of distinct balls into a set of distinct boxes, if no boxes can ...
0
votes
0answers
17 views

How do we deal with such arrangement problems?

15 different balls are kept in a straight line. Then their order is changed such that no ball is adjacent to a ball which it was adjacent to earlier. In how many ways can this task be ...
2
votes
0answers
27 views

What is the motivation behind the study of pattern-avoiding permutations?

There is a ton of research on pattern-avoiding permutations (permutations that do not contain some designated permutation pattern). We're figuring out how to enumerate them, what random ones are ...
0
votes
0answers
18 views

How many arrangements (correct and incorrect) possible with cube puzzle pieces in a particular unfolded form?

I am trying to do a computer program for arranging cube puzzle pieces in an unfolded form. A cube can be unfolded into 11 forms or nets. I have chosen a single net for now for simplicity purposes and ...
1
vote
2answers
44 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
0
votes
2answers
40 views

Differentiating between rearrangements and permutations.

I came across the text given below from the book Combinatorics - a problem oriented approach: Binomial Expansions If we expand the ...
3
votes
2answers
258 views

rearrangements of ABCDE having exactly 1 letter in its original position.

The question asks to find the no. of rearrangements of ABCDE having exactly 1 letter in its original position. (Rearrangement of a set means any arrangement of the set including its original ...
0
votes
1answer
56 views

Number of binary strings of length 8 that contain either three consecutive 0s or four consecutive 1s

How many bit strings (Consists of only 0 or 1) of length 8 contain either three consecutive 0s or four consecutive 1s? I am getting answer 256 but the provided answer is 147 . Can anyone explain ...
0
votes
1answer
24 views

Is there a way to count the number of unique possible songs able to be created?

As in the number of permutations of musical notes, lyrics (all known languages), and so on? I am no mathematician or musician so this might (probably) be totally stupid. Any ideas would be ...
1
vote
1answer
33 views

Number of distinct colourings for the regular pentagon

This is an answer check for the number of distinct colouring's for the regular pentagon given only four colour choices. I have the rotational group action ...
1
vote
1answer
44 views

permutation $\pi$, type, permutation $\sigma^4 = \pi$

Permutation $ \pi$ has a signature $2^43^5$. Find number of permutation $\sigma$ such that $\sigma^4 = \pi$ Could you give me a clue ?
0
votes
2answers
93 views

Ways to place 7 balls in 14 boxes.

How many ways are there to place 7 balls in 14 boxes. Balls are numbered from 1 to 7. One box can contain only one ball. And out of 14 boxes atleast 6 boxes must contain first 6 balls. 7th ball is ...
0
votes
1answer
37 views

Permutation_and_combinations_basic [closed]

In How many ways can we arrange numbers from 1 to 7 in n boxes such that we can repeat any number any number of times?But all the numbers from 1 to 7 must at least appear once. for example - If n = 7 ...
2
votes
4answers
43 views

Simple combinatorics dice problem

How would you explain, how ${5\choose4}$ corresponds to the pattern $6-6-6-6-3$ when throwing five dice. The pattern is meant to form a sum of $27$ in total. What I do understand is that you could ...
1
vote
1answer
25 views

What is the number of strings satisfying the following constraints?

We need to make a string of size $n$. Each character of the string is either ‘R’, ‘B’ or ‘G’. In the final string there needs to be at least r number of ‘R’, at least b number of ‘B’ and at least g ...
0
votes
1answer
59 views

The number of ways people standing in a line can be holding hands

I'm writing a program to analyze the maximum unique sequences of data in a string, given certain sets of two can be interpreted in two ways. There's a bit of math that I can't figure out, I've ...
2
votes
2answers
54 views

Combinations: People want a beer, there are certain kinds of beer, but limited numbers of each kind

Four people go to a pub and each wants to drink a pint of either the lager, ale, or porter. However, there are only 2 pints of lager, 1 ale, and 1 porter available to drink. How many combinations of ...
2
votes
0answers
59 views

Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $). Looking at 3-digit ...
1
vote
3answers
74 views

5-letter strings using the letters in the word “EVERGREEN”

From the word EVERGREEN, 5 letters are chosen at random and arranged into a string of letters. What is the probability that this string is palindromic?
3
votes
1answer
52 views

Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = ...
1
vote
0answers
41 views

Simple König theorem

I have to prove the "simple" König theorem, without using the marriage theorem: Let $S$ be a set of size $mn$. Suppose that $S$ is partitioned into $m$ subsets, all having size $n$, in two ways: ...
1
vote
3answers
47 views

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same.

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same. In other words, putting a book in space 1 and a book in ...
0
votes
0answers
31 views

Given a particular order how many times will it appear in all the possible permutations it has?

I have $10$ different coloured balls. I'm interested in selling them in packs of $15$ and the order is important. I know there are $10^{15}$ different ways of arranging these balls if I include the ...
0
votes
1answer
30 views

Number of ways to divide variables into two categories

I'm looking for a possible solution to find out the maximum number of combinations that can be derived from the given variables. If I'm not mistaken, I think permutations and combinations is the way ...
0
votes
1answer
57 views

distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
0
votes
0answers
30 views

What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?

I worked out the following expression as the number of all possible "words" consisting of exactly $w$ letters from an alphabet $L$ of size $\left|L\right| = n \leq w$, and containing each of these $n$ ...
0
votes
1answer
31 views

Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
2
votes
2answers
65 views

Question of Permutation and combination

I have found a question from somewhere in the internet as follows: English language has 26 alphabets, out of 4 distinct vowels and 7 distinct consonants, how many letter patterns can be made ...
6
votes
1answer
33 views

Equivalent of a sequence in regard to a certain length of a cycle for $\mathfrak{S}_{n}$

Let $n \in \Bbb{N}$ ( for me $0\notin \Bbb{N})$. Find the limit as $n$ tends to $+ \infty$ of the following sequence $$\frac{\alpha_{n}}{n}$$ where $\alpha_{n}$ is the number of permutations of ...
1
vote
3answers
51 views

Number of license plates formed by four digits and one letter, qualified.

I need some help with this question: If a license plate for a vehicle consist of five characters: $4$ digits (the first of which cannot be $0$), followed by one letter of the alphabet (which ...
9
votes
3answers
128 views

Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
2
votes
0answers
79 views

permutation and combination advanced

I have n sets having values less than 100. I need to find how many arrangements could be made if I pick one element from each set such that in the given arrangement there are no duplicates? NOTE: A ...
3
votes
3answers
38 views

Number of $r$ letter words taking letters from a $n$ letter word

I can't figure out how to do questions such as this one, any thoughts? What is the number of four letter words that can be formed from the letters in BUBBLE ...
1
vote
2answers
94 views

Seemingly simple combinatorial problem

Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?
1
vote
5answers
70 views

Permutations of $9$ balls of $3$ colors [closed]

I have $9$ balls total, $3$ red balls, $3$ green balls and $3$ blue balls. How many ways I can arrange them?
0
votes
1answer
62 views

Number of ways by which we can form a n digit number such that no two digit are same in the number?

Example : 2 digit number : so all two digit number except 11 22 33 44 55 66 77 88 99...this is simple but how to generalize for a number of n digit?(Also at each place any digit from 0 to 9 can come ...
0
votes
3answers
48 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 ...
0
votes
0answers
46 views

Number of possible ways to give change for 2 pounds

I was solving the following question on Project Euler In England the currency is made up of pound, £, and pence, p, and there are eight coins in general circulation: 1p, 2p, 5p, 10p, 20p, ...
0
votes
0answers
55 views

The probability of random permutation leaving the sequence almost unchanged

So let's say I have $52$ completely different kinds of arenas that get shuffled. What are the chances of getting the exact same sequence if only one arena can be out of order? For example, you could ...
1
vote
2answers
82 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 ...