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Recent questions in Probability and combinatorics

nar6jetaime86 2022-09-05

The answer to our problem ( $293$ ) is the coefficient of $x^{100}$ in the reciprocal of the following:
$(1 - x) (1 - x^{5}) (1 - x^{10}) (1 - x^{25}) (1 - x^{50}) (1 - x^{100})$

Cindy Noble 2022-09-04

Produce a set $A$ such that $r (n) > 0$ for all $n \in [1, N]$ , but with $| A | \leq \sqrt{4 N + 1}$ .
Note that
$r (n) = | {(a, a^{'}) : a, a^{'} \in A, n = a + a^{'}} |$
$A = {0, 1, 2}$ would work with the interval being $[1, 4]$ . Then $3 \leq \sqrt{17}$ .
A second part of the question shows that one can prove that $| A | \leq \sqrt{N}$ if it satisfies the above conditions. But $3 > \sqrt{4} = 2$ . Does this mean that my set $A$ is wrong?

hifadhinitz 2022-09-04

How many squares of all sizes arise using an 𝑛-by-𝑛 checkerboard? How many triangles of all sizes arise using a triangular grid with sides of length 𝑛 ?

ubumanzi18 2022-09-04

Let's $G = (U, V, E)$ be a balanced bipartite graph which $| U | = | V | = n$ and $| E | = n * (n - 1)$ ; All nodes in $U$ are connected to all nodes in $V$ except $u_{i}$ to $v_{i}$ for $1 \leq i \leq n$ .
Definition 1: Cross edges are two edges in $E$ , one with two end points $u_{i}$ , $v_{j}$ and the other with $u_{j}$ , $v_{i}$
Definition 2: Good-perfect matching is a perfect matching with no cross edges.
What is the complexity of counting the number of Good-perfect matching in $G$ ?

oliadas73 2022-09-03

Let $k \geq 1$ be fix and $b_{n}$ be the amount of possible words $w = v_{1} \dots v_{n}$ of length $n$ on the alphabet ${1, \dots, k}$ , such that $v_{i} \neq v_{i + 1}, 1 \leq i \leq n - 1$ .
a) Show by counting that
$b_{0} = 1 and b_{n} = k (k - 1)^{n - 1} for n \geq 1.$ .
b) Identify the generating function $\sum_{n \geq 0} b_{n} x^{n}$
My try:
a) first. For the first element of each word there are $k$ possibilities. For every successor there are $(k - 1)$ possibilities because they depend on the element before themselves.
Is this correct and complete?

ubwicanyil5 2022-09-03

A marble is selected at random from ajar containing 4 red marbles, 3 yellow marbles, and 6 green marbles.
What is the probability that the marble is red?

chrisysakh 2022-09-03

There are two boxes: Box A and Box B. Box A contains 5 statistic books and 3 calculus books. Box B contains 4 statistic books and 2 calculus books. A coin is thrown twice. If at least 1 Head appears, a book from box A will be taken. Otherwise, a book from box B will be taken. a. If 2 books are taken, what is the probability that those are calculus books?

tsuyakas1 2022-09-02

A pair of dice is rolled. What is the probability of getting
a)a sum greater than 3?
b)a sum less than 5?
What is the probability of getting a sum greater than 8?
What is the probability of getting a sum less than 5?

Cristal Travis 2022-09-02

if 3 people are randomly selected, what is the probability that they are all born in March?

Zaiden Soto 2022-08-20

Playing the lottery. New York State’s “Quick Draw” lottery moves right along. Players choose between 1 and 10 numbers from the range 1 to 80; 20 winning numbers are displayed on a screen every four minutes. If you choose just one number, your probability of winning is 20/80, or 0.25. Lester plays one number eight times as he sits in a bar. What is the probability that all eight bets lose?

polynnxu 2022-08-18

How many possible 10-digit phone numbers with the area code 548 are there? (any decimal digit can be the digit of a phone number. For instance the number 548−050−5000 is a possible phone number)

metodystap9 2022-08-17

$Find the probability P = (0 < z < 1.67), using the standard normal distribution 42.25 % 45.54 % 42.07 % 35.54 %$

ljudskija7s 2022-08-09

There are 8 area code numbers in the Boston area $(351, 978, 617, 857, 339, 781, 508, 774)$ . If a telephone number is chosen at random explain why the probability of if the number to start with 617 is not 1/8.

tuanazado 2022-08-05

(12x - 3)degrees + (10x + 15) degrees

Tarnayfu 2022-08-02

Use the Euclidean algorithm to find gce(34,126) and write it as a linear combination of 34 and 126.

capellitad9 2022-07-23

Finding a number that is smaller than 100 and that has more factors than 100.

comAttitRize8 2022-07-16

How many different phone numbers are possible in the area code 503, if the first number cannot start with a 0 or 1

doturitip9 2022-07-15

Choose numbers from $1$ to $2 n$ uniformly at random. How many numbers must be chosen, on average, before at least $n$ numbers have been picked?

grenivkah3z 2022-07-14

Let $e (n)$ be the number of partitions of $n$ with even number of even parts and let $o (n)$ denote the number of partitions with odd number of even parts. In Enumerative Combinatorics 1, it is claimed that it is easy to see that $\sum_{n \geq 0} (e (n) - o (n)) x^{n} = \frac{1}{(1 - x) \times (1 + x^{2}) \times (1 - x^{3}) \times (1 + x^{4}) \times . . .}$ . I have been racking my head over this for the past few hours, and I can't see any light.
Noticed, that $e (n) - o (n) = 2 e (n) - p (n)$ where $p (n)$ is the number of partitions of $n$ , so the above claim is equivalent to showing $\sum_{n \geq 0} e (n) x^{n} = \frac{1}{2} \frac{1}{(1 - x) (1 - x^{3}) (1 - x^{5}) . . .} (\frac{1}{(1 - x^{2}) (1 - x^{4}) . . . .} + \frac{1}{(1 + x^{2}) (1 + x^{4}) . . . . . . . .})$ , and similarly, it is equivalent to $\sum_{n \geq 0} o (n) x^{n} = \frac{1}{2} \frac{1}{(1 - x) (1 - x^{3}) (1 - x^{5}) . . .} (\frac{1}{(1 - x^{2}) (1 - x^{4}) . . . .} - \frac{1}{(1 + x^{2}) (1 + x^{4}) . . . . . . . .})$ , but these identities appear more difficult than the original one.

Janessa Olson 2022-07-14

Simple combinatorics and probability theory related question
5 apples are randomly distributed to 4 boxes. We need to find probability that there are 2 boxes with 2 apples, 1 box with 1 apple and 1 empty box.
I'm getting the correct answer with $\frac{\frac{5!}{2! 2! 1! 0!} * 4 * 3}{4^{5}} = 0.3515625$ (anyway, the answer is said to be 0.35, but I think it is a matter of rounding).
But I don't understand why there are $4^{5}$ elementary events in total. Firstly, I thought It should be $((\binom{4}{5}))$ - number of combinations with repetitions, but I couldn't get the proper answer.
Isn't approach with $((\binom{4}{5}))$ elementary events more correct?

Even though it is always placed into the field of Precalculus, probability and combinatorics problems are more related to analytics and statistical information that many students implement as they are working with the accuracy of their research data. It is one of the reasons why finding good probability combinatorics examples is both complex and not. It is good to have a look through probability combinatorics questions as these will provide you with the answers and help understand how equations work and why possible outcomes are vital for understanding how the calculation takes place.

Precalculus

Combinatorics: Examples, Equations, and Practice Problems