# Tagged Questions

The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.

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### How would I figure out how many anagrams of mississippi don't contain the word psi?

I'm really confused how I'd calculate this. I know it's the number of permutations of mississippi minus the number of permutations that contain psi, but considering there's repetitions of those ...
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### bijection = bijection + bijection on symmetric integer intervals

Given a bijection $f:\mathbb Z \to \mathbb Z$ where $\mathbb Z$ is the set of all integers, does there always exist two bijections $g:\mathbb Z \to \mathbb Z$ and $h:\mathbb Z \to \mathbb Z$ which ...
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### Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
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### Which graphs can be drawn using straight lines with no disjoint edges?

What is the class of graphs that can be drawn using only straight lines with no two edges disjoint? Edges are disjoint when they don't cross and they don't share a vertex. Vertices should be in ...
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### A group acting on functions of functions of functions

Given a group acting on a set $X$, there is a standard way to define an action of the group on the set of functions of $X$. This can be extended to the set of functions of functions of $X$ as I show ...
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### How many distinct functions can be defined from set A to B?

In my discrete mathematics class our notes say that between set A (having 6 elements) and set b (having 8 elements), there are $8^6$ distinct functions that can be formed, in other words: $|b|^{|a|}$ ...
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### Largest prime factor of 600851475143 [duplicate]

I'm trying to use a program to find the largest prime factor of 600851475143. This is for Project Euler here: http://projecteuler.net/problem=3 I first attempted this with the code that goes through ...
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### How to prove indirectly that if $42^n - 1$ is prime then n is odd?

I'm struggling to prove the following statement: If $42^n - 1$ is prime, then $n$ must be odd. I'm trying to prove this indirectly, via the equivalent contrapositive statement, i.e. that if $n$ ...
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### Help with combinatorial proof of identity: $\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$

How to prove this identity? Can someone please give me some insight ? $$\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k} \binom{n}{k} = \sum_{k=1}^{n} \frac{1}{k}$$
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### What is the difference between necessary and sufficient conditions?

If $\quad p \implies q\quad$ ($p$ implies $q$), then $p$ is a sufficient condition for $q$. If $\quad \bar p \implies \bar q \quad$ (not $p$ implies not $q$), then $p$ is a necessary condition for $q$...
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### Fibonacci numbers divisible by $9$

The $n$th Fibonacci number $F_n$ is defined as follows,$$F_1=F_2=1\mbox{ and } F_{n+2}=F_{n+1}+F_{n}\mbox{ for } n\geq 1$$ I want to know how many of the first $1000$ Fibonacci numbers are divisible ...
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### What exactly is the difference between weak and strong induction?

I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching the $k^{th}$ rung of a ladder and ...
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### Proof by Induction: Alternating Sum of Fibonacci Numbers [duplicate]

Possible Duplicate: Show that $f_0 - f_1 + f_2 - \cdots - f_{2n-1} + f_{2n} = f_{2n-1} - 1$ when $n$ is a positive integer This is a homework question so I'm looking to just be nudged in the ...
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### Minimum degree of a graph and existence of perfect matching

I was reading a result where the following proposition appears as a preliminary step (and left as exercise): Claim: Suppose $G$ is a graph on $n$ vertices ($n$ even and $n \geqslant 3$) with ...
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### Number of ways to write $n$ as sum of odd or even number of Fibonacci numbers

In our discrete mathematics exercises I came of with the question: Prove that the coefficients of $\prod_{n\geq2}{(1-x^{F_n})}=1-x-x^2+x^4+x^7+\dots$ can only be $-1,1$ or $0$, where $F_n$ ...
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### Show by combinatorial argument that ${2n\choose 2} = 2{n \choose 2} + n^2$

So i was given this question. Show by combinatorial argument that ${2n\choose 2} = 2{n \choose 2} + n^2$ Here is my solution: Given $2n$ objects, split them into $2$ groups of $n$, $A$ and $B$. $2$-...
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### Interesting shapes using probability and discrete view of a problem

Suppose we have a circle of radius $r$, we show the distance between a point and the center of the circle by $d$. We then choose each point inside the circle with probability $\frac{d}{r}$ , and turn ...
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### Algorithm to compute fastest method of collecting $k$ re-spawning items which spawn at $n$ specified points

Let $V = v_1, \dots, v_n$ be the locations the items can spawn at, and let $U = u_1, \dots, u_k$ be the current positions of the items. We will assume a new items spawns instantly every time we ...
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### Binomial formula for $(x+1)^{1/3}$ (related to Newton's binomial theorem)

I know that $$\displaystyle \sqrt{1+x} = \sum_{j=0}^{\infty}\left( \frac{(-1)^{(j-1)}}{2^{2j-1}\cdot(2j-1)}\binom{2j-1}{j}x^j\right).$$ Now, I want to evaluate $\sqrt[3]{1+x}$ but stuck at some ...
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### Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful ...
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### prove for all $n\geq 0$ that $3 \mid n^3+6n^2+11n+6$

I'm having some trouble with this question and can't really get how to prove this.. I have to prove $n^3+6n^2+11n+6$ is divisible by $3$ for all $n \geq 0$. I have tried doing $\dfrac{m}{3}=n$ and ...
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### If $xy$ and $x+y$ are both even integers (with $x,y$ integers), then $x$ and $y$ are both even integers

The title statement can be proven using the contrapositive, note that $x$ odd or $y$ odd means that at least one of $x\cdot y,x+y$ is odd. Is there a way to prove the statement directly? To ...
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### How to distinguish between combination and permutation questions?

How do you distinguish combination and permutation question? An example of a combination question: Example: How many different committees of 4 students can be chosen from a group of 15? ...
How many unique integers can you get from $\lceil2012/n\rceil$ where $n$ is a positive integer? I don't know at all where to begin to approach this problem. I thought it maybe had something to do ...
You are playing a game, your goal in this game is to catch a frog that's leaping between natural numbers. At first, the frog is found at the number $a \in \mathbb N$ which is not known to you. Each ...