I’m not exactly sure how to do this using mathematical induction. Thanks for the help. I’m trying to prove that the product rule is valid for some number of tasks $m$, such that $m$ is greater than two using induction and the product rule for two tasks as a given. The product rule for a […]

Let’s say we are flipping a coin $n$ times. What is the probability that we always have more heads than tails. Meaning that if we are counting the number of times we have had heads and the number of time we have had tails, what is the probability that throughout the $n$ throws we continuously […]

Given a graph $G=(V,E)$, the line graph of $G$ is a graph $\Gamma$ whose vertices are $E$ (the edges of $G$) and in $\Gamma$, two vertices $e_1,e_2$ are connected if, as edges in $G$, they share an endpoint. Now let $G$ be the $3D$ cube graph. It has $8$ vertices, $12$ edges and it is […]

How many four-sided figures appear in the diagram below? I tired counting all the rectangles I could see, but that didn’t work. How do I approach this?

Given a number, say $x$, and a set of numbers made up of only $k$ different numbers, where each of the $k$ numbers is repeated $n_1,n_2,\dots n_k$ times. How do I tell if it is possible to find a subset such that it sums to $x$. E.g.: $$x=6 , k=3$$ $$S=\{1,2,3,3,2,1\}$$ $$n_1=2, n_2=2, n_3=2$$ $$1+2+3=6$$ […]

How many numbers must be selected from the set $\{1,2,3,4,5,6,7,8,9\}$ to guarantee that at least one pair of these numbers add up to $10$? Justify your answer. Here’s my answer. Consider the two sets $\{1,2,3,4\}$ and $\{6,7,8,9\}$. In the worst case, one may pick the number $5$ and then all the numbers from one set […]

During a dinner with $k=20$ persons sitting at $n=4$ tables with $m=5$ seats, everyone wants to share a table with everyone. The assembly decides to switch seats after each serving towards this goal. What is the minimal number of servings needed to ensure that everyone shared a table with all the other persons at some […]

Given $n$ distinct elements, how many Young Tableaux can one make?

I need to find a function to enumerate the ordered list of sequential words based on a charset. Let me give you an example. If the charset is “abc”, the function to be found “f” should compute the following: f(0) = a f(1) = b f(2) = c f(3) = aa f(4) = ab f(5) […]

This is similar to my previous question, Number of 5 letter words over a 4 letter group using each letter at least once. The only difference is that there are 3 letters to choose from instead of 4. However, I’ve run into a problem. Using inclusion exclusion I get: $3^5 – 3 \cdot 2^5 + […]

Intereting Posts

Most natural intro to Complex Numbers
Taking trace of vector valued differential forms
Finding the general solution to a system of differential equations
How to find the value of $\int_0^1 \frac{x-1}{\log x}\,dx$?
What is the *middle* digit of $3^{100000}$?
Are squares of independent random variables independent?
Give a connected graph whose automorphism group has size 3
References for Topology with applications in Engineering, Computer Science, Robotics
Quotient topologies and equivalence classes
Is this space contractible?
Are there infinitely many natural numbers $n$ such that $\mu(n)=\mu(n+1)=\pm 1$?
Spectrum of infinite product of rings
Obtaining binomial coefficients without “counting subsets” argument
Comparing Category Theory and Model Theory (with examples from Group Theory).
Limit of an integral (remainder term of a Euler-Maclaurin expansion)