0
votes
2answers
14 views

Probability of weather on consecutive days.

Probability of a cloudy day is .55 Probability of a sunny day is .45 A)What is the probability of three consecutive cloudy days, followed by a sunny day? B)What is the probability that exactly 1 out ...
-1
votes
0answers
11 views

Using BCR experiment [on hold]

consider a random experiment of observing a mechanical or electrical unit consisting of five components and determining which components are working and which have failed. Use the BCR to find the ...
-1
votes
2answers
38 views

In tossing 5 6-sided fair dice, what is the probability of at least one 2 if the dice are indistinguishable?

I know that the answer is .4 because it is given. I just do not know how to get there. The answer would be .598 if the dice were distinguishable (ordered).
0
votes
1answer
55 views

Number of surjective functions from $\{1,2,…,n\}$ to $\{a,b,c\}$

Ok so following questions are given in my text book Let $A = \{1, 2, 3,...., n\}$ and $B =\{a, b, c\}$ then the number of functions form $A$ to $B$ that are onto is. I have no idea how to find ...
-3
votes
1answer
25 views

Generating Functions [on hold]

Write a generating function that models the number of distributions of r identical balls into 5 distinct boxes if each box has between two and seven balls. Then find the coefficient that gives the ...
1
vote
3answers
75 views

Integer solutions Help

How many integer solutions are there to $$x_1 +x_2 + \text{ ... }+x_5 =31 \;\; \text{ with } \; \; x_i \geq i, \;\; i=1,2,3,4,5$$ I tried it and got $C(20,16)$ but I don't really think that is ...
1
vote
1answer
28 views

Possible arrangments Letters?

How many arrangements are possible of the letters in EZPZ I CAN DO IT, which has five vowels (A, E, I, I, O) and seven consonants (C, D, N, P, T, Z, Z). a) if there are no restrictions, b) if ...
1
vote
2answers
26 views

How do I calculate variance for sum of dice?

I'll post my work, but I'm not sure how to calculate variance. The question asks for the expected sum of 3 dice rolls and the variance. I think I got the expected sum. Any help would be awesome :) ...
0
votes
1answer
39 views

Probability: put 20 distinct balls randomly in 12 urns

You put 20 distinct balls randomly into 12 urns. What is the probability of having 3 urns with 4 balls each and 4 urns with 2 balls each (the other 5 urns are left empty). For my sample space I have: ...
0
votes
1answer
13 views

Simple question about choosing items from a box

Let's say I have a box of twelve balls and eight are blue and the rest red. When I choose seven balls at random, what is the probability of getting exactly two blue balls? I know it's a fraction ...
1
vote
6answers
211 views

Prove that $\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$

Prove that: $$\sum_{k=0}^n k^2{n \choose k} = {(n+n^2)2^{n-2}}$$ i know that: $$\sum_{k=0}^n {n \choose k} = {2^n}$$ how to get the (n + n^2)?
1
vote
1answer
23 views

What's wrong with this recursion of counting codes of length $n$ formed by $a$, $b$, and $c$ such that no three consecutive letters are distinct

I found the following problem in a combinatorics book and gave it a try. Let $B_n$ denote the set of codes of length $n$ formed by using the letters $a$, $b$, and $c$, none of which contains three ...
2
votes
1answer
56 views

Algebraic Combinatorics

Let $K_{r,s}$ denote the complete bipartite graph, defined on $r + s$ vertices $\{v_1,v_2,...,v_r,w_1,...,w_s\}$, with an edge between $v_i$ and $w_j$ for $1 ≤ i ≤ r$ and $1 ≤ j ≤ s$. By ...
0
votes
1answer
26 views

Suppose we toss a coin 5 times and define Y as number of runs of heads. How do you find expectation and variance?

Range would be {0,1,2,3,4,5} Is there an easy way to find expectation rather than writing out all the possible outcomes?
1
vote
0answers
25 views

Finite sum identity involving Stirling numbers

I was given the following identities (note: $s_{n,k},S_{n,k}$ are the Stirling numbers of first and second kind respectively): (1) $s_{n+1,k+1}=\sum_{i=k}^{n}\binom{i}{k}s_{n,i}$ (2) ...
0
votes
1answer
17 views

Number of ways of arranging 7 coloured blocks in patterns

The question goes: Given you have 7 differently coloured blocks (red, orange, yellow, green, blue, indigo, violet), how many different arrangements are there such that the blue and green are not ...
1
vote
1answer
20 views

Placing between n and n+2 different books into n different boxes

The homework question asks: There are n (for integers $n>1$) different boxes, each of which can hold up to $n+2$ books. Find the probability that: a) No box is empty when $n$ different ...
4
votes
1answer
74 views

Proving a combinatorial identity “directly”

This is a homework problem. In the first part of the problem, I managed to use a combinatorial problem to prove the following identity: $\Sigma_{k=0}^{n}(-1)^k {2n-k \choose k} 2^{2n-2k} = 2n+1$ But ...
0
votes
2answers
45 views

Two sequences $a$ and $b$ for which $\Delta a_n = \Delta b _n$

Find two different sequences $a$ and $b$ for which $\Delta a_n = \Delta b_n$ for all of $n$. This is my first time doing recurrence relations, so if anyone could provide some thorough and clear ...
0
votes
2answers
36 views

Different values of $x$ and $y$ between $\sqrt{39}$ and $\sqrt{224}$

If $x$ and $y$ are whole numbers between $\sqrt{39}$ and $\sqrt{224}$, then how many different values can $x$ + $y$ have? OK, first I found that the set numbers are: $$7, 8 ,9 ,10 ,11 ,12, 13,14$$ ...
0
votes
4answers
31 views

Finding nth term application problem

I was given this question class today and I wasn't quite sure how to solve it "There are $10$ computers all connected with a cable to each other computer" 1) How many wires are there? 2) How many ...
1
vote
2answers
46 views

Coefficients of this generating function

For the first part of a problem, I solved the generating function to be $F(x) = \frac{x^3}{(1-x)^2}$ Now it's the easy part that has me a little confused. What would the coefficients be in this case? ...
0
votes
0answers
53 views

Alternating permutation exponential generating function

A permutation pi is alternating if pi_1 > pi_2 < pi_3 > pi_4 <….Let a(n) be the number of alternating permutations of size n. (a) Find a recurrence relation for a(n). (b) Evaluate the ...
0
votes
1answer
74 views

Round table exponential generating function

Let $r(n)$ be the number of different ways to seat $n$ people around a round table. Find the exponential generating function for $r$. I believe $r(n)$ is just equal to $n!/n = (n-1)!$. So then I ...
1
vote
1answer
26 views

Probability: 30 balls in a bucket, homework

i need some help with some homework, first time i am doing probability and statistics, id like to know if my 2 answers below are correct, and how i can solve the remaining 2. There are 30 balls in a ...
1
vote
0answers
34 views

How to answer the following question related to counting the number of trees of a graph?

I am asked to prove the equality $$ 2(n-1)n^{n-2} = \sum_{k=1}^{n-1} \binom{n}{k} k(n-k)T(k)T(n-k) , $$ where $T(k)$ is the number of different trees with $k$ numbered vertices. I think the ...
0
votes
2answers
51 views

Problem with Set Theory Counting Principle

I'm trying to apply the counting principle to the following: "Of 300 people: 35 - bicycle and car. 40 - car and bus. 60 - bicycle and bus. 90 - bicycle. 70 - car. 105 - bus. 25 - bicycle, car, and ...
1
vote
2answers
56 views

Show how the probability that an 8 character password contains exactly 1 OR 2 integers is .630

A password is 8 characters long. Each character can contain 26 lower case or 26 uppercase letters or a integer from 0-9. What is the probability that an 8 character password contains exactly 1 OR 2 ...
1
vote
1answer
180 views

How many n-digit numbers contain at least one 2 and at least one 3, but no 7’s?

Additional rule: each digit is one of $\{0,1,2...9\}$ and the first digit is nonzero. I think this question isn't hard, I just don't seem to be clear about the question. My interpretation: How many ...
1
vote
2answers
171 views

Number of ways to form a 3-letter word with repetition allowed?

The additional rule is: no letter can be used more often than it appears in MILLENNIUM? (Which is pretty logical I guess) MILLENNIUM = MM, II, LL, NN, E, U My logic: Case 1: Double letters + 1 ...
0
votes
2answers
142 views

No. of 5-digit monotonic numbers

The monotonic number is made of digits 1, 2, …, 9, such that each subsequent number equal to or greater than the previous number. Examples: 11119, 12369, 18999 etc. I understand that I can isolate ...
0
votes
3answers
72 views

Forming words from letters

If we have five letters e.g. a,b,c,d,e a. How many four-letter words can we make that have exactly two vowels and two consonants? b. from (a), how many of those words have distinct vowels?
0
votes
4answers
105 views

Six Boys, Six Girls. How many possible partners?

I'm struggling to understand the solution to this question concerning combinatorics. Question: At a party six boys and six girls dance together. Assuming that classical dance is performed, in which ...
1
vote
2answers
67 views

Generating functions for compositions

Let $g(n)$ be the number of compositions of n where each part is an odd number. Let $h(n)$ number of compositions of $n$ where each part is either 1 or 2. Using the ordinary generating functions ...
2
votes
2answers
42 views

A man, woman, boy, girl, cat, and dog are walking down a path..

I'm hoping someone can explain how this works. The problem: A man, woman, boy, girl, cat, and dog are walking down a path in single file. How many ways can this happen if the dog is between the man ...
0
votes
0answers
17 views

The summation of product of factorials

So the question is $\sum\limits_{x=0}^n \frac{(\beta+n-x)! (\alpha+x)!}{x!(n-x)!}$. I got the following result from mathematica yet I don't know how to prove it. Can anyone give me some help?
1
vote
2answers
80 views

Binomial Expansion.

So I had a question: Prove that for $n \geq 1$, $${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...+ n{n \choose n} = n2^{n-1}$$ So my idea was to take the binomial expansion of $(1+1)^n$ which ...
1
vote
2answers
51 views

Algebraic proof for the following identity

Give an algebraic proof that $\binom{n+1}{m+1} = \sum_{k=m}^{n} \binom{k}{m}$. I've tried using Pascal's rule and looking for a telescopic sum, but I can't find one. Any help is appreciated.
2
votes
2answers
27 views

Ordinary Generating functions for $b_n$

Problem Let $f(x)$ be a ordinary generating function for the sequence $ \{\ a_0, a_1, a_2... \}\ $ Find the ordinary generating function for $b_0 = b_1 = 0, b_2 = 1$ $b_n = a_n$ for $n \geq 3$. Also ...
2
votes
1answer
26 views

Some help with generating functions

Problem Let $f(x)$ be the ordinary generating function for the sequence $ \{\ a_0 , a_1, a_2,... \}\ $. Find the ordinary generating sequence for the following sequence: $$b_n = a_n + c \ \ \ , n \in ...
3
votes
1answer
38 views

Simple combinatorics question, sanity check.

Sorry if I'm wasting everyone's time. I'm checking a homework set, and I'm a little confused by the solution manual's answers, and want a second opinion before shrugging it off as a mistake. Consider ...
0
votes
2answers
36 views

Ordinary generating functions problem

Problem Find the ordinary generating function for each of the following sequences. In each case the sequence is defined for all $n \in \mathbb{N}_0$. $$a_n = n$$ I'm having a very hard time ...
2
votes
1answer
56 views

Compositions - Fruit Salad

I'm asked to find $s(n)$ which is the number of ways to make a fruit salad with $n$ pieces of fruit, given that we must use strawberries by the half-dozen, an odd number of apples, between 2 and 7 ...
1
vote
1answer
66 views

You bought six numbers at your local hardware store. The numbers are 0, 1, 2, 3, 4, 5.

I got this question and can't crack it. Any help will be appreciated. You bought six numbers at your local hardware store. The numbers are 0, 1, 2, 3, 4, 5. a) How many 6 digit house numbers would ...
1
vote
1answer
44 views

Combinatorics with balls and bins with constraint

I have $90$ identical balls to distribute among $64$ distinguishable bins. Each must get at least 1 ball, after that the distribution over the remaining $26$ does not matter. I know I first have to ...
3
votes
1answer
43 views

Number of subsets without consecutive numbers

Consider $S=\{1,2,\ldots,15\}$. Let $X$ denote the number of subsets of $S$ of four elements which contain no consecutive numbers. The claim is that $X$ equals the coefficient of $x^{14}$ in ...
2
votes
1answer
31 views

Counting problem involving Hat Check experiment with n hats

The question is a spin on the Hat Check problem. "There are n= 2k hats (an even number). Find the probability of B = { $h_{i} = i$ if $i$ is even and $h_{i} \neq i$ if $i$ is odd)." My interpretation ...
4
votes
3answers
84 views

How to answer the following combinatorial question?

Consider the set $S = \{ 1,2, \cdots , 15 \}$. The general question is: how many subsets of $S$ exist such that the subsets contain 4 different elements without any consecutive numbers? As a ...
0
votes
2answers
23 views

get a 'win' in even expriment

unfair coin give '$x$' in probability $p$ and '$y$' in ($1-p$). we toss it up until we get '$x$'. what is the probability that we get '$x$' in even tossing? I know that for even tossing we need ...
0
votes
0answers
29 views

Possible number of combinations for a shape of cubes

My Introduction to Engineering teacher gave us the following exercise: List all of the possible combinations of shapes that you can make with cubes, with the whole thing being no more than 3 x 3 x 3 ...