The inclusion-exclusion principle states that the number of elements in the union of two given sets is the sum of the number of elements in each set, minus the number of elements that are in both sets.
Questions tagged [inclusion-exclusion]
1552 questions
88
votes
10 answers
100 Soldiers riddle
One of my friends found this riddle.
There are 100 soldiers. 85 lose a left leg, 80 lose a right leg, 75
lose a left arm, 70 lose a right arm. What is the minimum number of
soldiers losing all 4 limbs?
We can't seem to agree on a way to approach…
68
votes
5 answers
Number of onto functions
What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements.
I found that if $m = 4$ and $n = 2$ the number of onto functions is $14$.
But is there a way to generalise this using a…
icyflame
- 935
47
votes
2 answers
What is the optimal number of dice to roll a Yahtzee in one roll?
Description
In the game of Yahtzee, 5 dice are rolled to determine a score. One of the resulting rolls is called a Yahtzee.
To roll a Yahtzee you must have 5 of a kind. (5 1's or 5 2's or 5 3's etc..).
In the game of Yahtzee you can only have 5…
Michael King
- 601
34
votes
6 answers
Why is the Derangement Probability so Close to $\frac{1}{e}$?
A derangement is a permutation $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and probability classes is to compute the number of derangements,…
Kevin P. Costello
- 6,538
23
votes
5 answers
In how many ways can $1000000$ be expressed as a product of five distinct positive integers?
I'm trying to solve the following problem:
"In how many ways can the number $1000000$ be expressed as a product of five distinct positive integers?"
Here is my attempt:
Since $1000000 = 2^6 \cdot 5^6$, each of its divisors has the form $2^a \cdot…
user144765
- 752
22
votes
1 answer
Is the Maclaurin series expansion of $\sin x$ related to the inclusion-exclusion principle?
When I see the alternating signs in the infinite series expansion of $\sin x$, I'm reminded of the inclusion-exclusion principle. Could there be any way to visualize it in such a way?
Also, is there an elementary reason why the Taylor series…
hollow7
- 2,545
21
votes
3 answers
Combinatorial argument to prove the recurrence relation for number of derangements
Give a combinatorial argument to prove that the number of derangements satisfies the following relation:
$$d_n = (n − 1)(d_{n−1} + d_{n−2})$$
for $n \geq 2$.
I am able to prove this algebraically but not able to see the combinatorial example.
Arjun Kaa
- 211
21
votes
5 answers
Inclusion-exclusion-like fractional sum is positive?
Let $A_1,A_2,\ldots,A_n$ be finite nonempty sets. Is it true that
$$\sum_{i=1}^n\frac{1}{|A_i|}-\sum_{1\leq i
nan
- 1,125
20
votes
5 answers
Generalised inclusion-exclusion principle
In answers to combinatorial questions, I sometimes use the fact that if there are $a_k$ ways to choose $k$ out of $n$ conditions and fulfill them, then there are
$$
\sum_{k=j}^n(-1)^{k-j}\binom kja_k
$$
ways to fulfill exactly $j$ of the conditions.…
joriki
- 242,601
19
votes
5 answers
Famous uses of the inclusion-exclusion principle?
The standard textbook example of using the inclusion-exclusion principle is for solving the problem of derangement counting; using inclusion-exclusion (and some basic analysis) it can be shown that $D(n)=\left[\frac{n!}{e}\right]$ which I consider…
Gadi A
- 19,839
19
votes
4 answers
Sum of binomial coefficients $\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$
How do I prove the following identity:
$$\sum_{k=0}^{n}(-1)^k\binom{n}{k}\binom{2n - 2k}{n - 1} = 0$$
I am trying to use inclusion-exclusion, and this will boil down to a sum like inclusion-exclusion, and the $\binom{2n-2k}{n-1}$ term wouldn't…
Mark
- 193
17
votes
1 answer
A general "inclusion-exclusion principle" / Formulas like $\inf(a,b)\sup(a,b)=ab$
Let me list a few formulas which should be well-known:
(GCD-LCM product formula) For positive integers $n,m$,
$$\operatorname{gcd}(n,m)\operatorname{lcm}(n,m)=nm.$$
(Inclusion-exclusion principle) For sets $A,B$,
$$\#(A\cap B)+\#(A\cup…
Luiz Cordeiro
- 19,213
- 35
- 67
17
votes
2 answers
Number of functions $f:\{1,...,n\}\to\{1,...,n\}$ that have $|f^{-1}(\{i\})|=i$ for some $i$
Let $S=\{1,...,n\}$, I am looking at number of functions functions $f:S\to S$ such that there exists $i \in S$ such that $$|f^{-1}(\{i\})|=i$$
I am guessing I am supposed to use PIE (Principle of Inclusion and Exclusion)
I let $X=\{f:S\to S\}$.
My…
Nasal
- 828
17
votes
2 answers
Arrangements of a,a,a,b,b,b,c,c,c in which no three consecutive letters are the same
Q: How many arrangements of a,a,a,b,b,b,c,c,c are there such that
$\hspace{5mm}$ (i). no three consecutive letters are the same?
$\hspace{5mm}$ (ii). no two consecutive letters are the same?
A:(i). 1314. ${\hspace{5mm}}$ (ii). 174.
I thought of…
Stoner
- 1,246
16
votes
1 answer
How many ways to arrange $n$ points in $(\Bbb F_q)^2$ with no three collinear?
How many ways are there to arrange $n$ points in the finite field plane $(\Bbb F_q)^2$ with no three of the points collinear?
An easy upper bound is $(q^2)^n=q^{2n}$, but of course it's less than that. (Of course, if I asked the same question over…
Akiva Weinberger
- 25,412