# Questions tagged [inclusion-exclusion]

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.

1,521 questions
Filter by
Sorted by
Tagged with
58 views

### $(w_{1},w_{2},w_{3},\dots,w_{7})$ integers with $20\le w_{i} \le 22$ such that $\sum_{i=1}^{7}w_{i} = 148$

How many $(w_{1},w_{2},w_{3},\dots,w_{7})$ where each of the $w_{i}$'s are integers and $20\le w_{1},w_{2},w_{3},\dots,w_{7}\le 22$ such that they satisfy $$w_{1}+w_{2}+w_{3}+\dots+w_{7}=148$$ ATTEMPT ...
• 189
85 views

### definition of cyclotomic polynomials

The $n$th cyclotomic polynomial can be expressed via the Mobius function as follows: $$\Phi_n(x) = \prod_{\substack{1\le d\le n\\d\mid n}}(x^d - 1)^{\mu(\frac{n}{d})}$$ In every reference I have ...
• 369
44 views

### Combinatorics questions, people in a row, i'th person not standing right to (i-1)th person [duplicate]

The question goes like: There are n people in a row, how many ways are there to rearrange them, such that for all $0 \le i \le n-1$, the people who stood in original (i+1)th place, is not standing ...
43 views

### Book of Proof chapter 3 section 7 (Inclusion-Exclusion Principle), exercise 4b. Need help finding the answer algebraically.

I've been stuck on Chapter 3 section 7 (Inclusion-Exclusion Principle) of the Book of Proof 3rd edition, in the exercises part, trying understand the logic behind the answer to question 4b: This ...
127 views

### Counting binary strings with $9$ ones and containing $11011$

I am currently trying to solve this problem: How many strings of 20 bits are there with exactly nine $1$s and containing at least one occurrence of $11011$ as a substring? I don't have problems with ...
51 views

### How many different 8 characters passwords with 2 upper-case 2 digits 4 lower-case

A web-banking password is always 8 characters long and it always comprises two upper-case letters from the standard English alphabet, two digits, and four lower-case letters from the standard English ...
• 254
54 views

68 views

### Application of the Principle of Inclusion/Exclusion and the Binomial Theorem in Combinatorial Proofs [closed]

Consider a set $Z=X \cup Y$, where $X=\left\{x_1, \ldots, x_n\right\}$ is a set of blue elements and $Y=$ $\left\{y_1, \ldots, y_m\right\}$ is a set of red elements. (a) How many subsets of $Z$ ...
• 195
1 vote
70 views

### Derangements for couples in a round table

Question: Let ( m(n) ) denote the number of ways of seating ( n ) married couples around a circle such that no husband sits next to his wife. Then, the remainder obtained on dividing ( m(5) ) by ( 5 ) ...
36 views

### Sanity Check Streak of Head's

You have a biased coin with probability $\frac{1}{3}$ of heads. If you flip the coin 10 times, what is the probability of having 8 or more heads in a row? I tried doing this question two ways, using ...
• 157
57 views

### Total permutations for n-married couples

Let A(n) denote the number of ways of seating n-married couples, around a circle, such that men and women sit alternately, and no husband sits next to his wife. Then Compute A(5): I tried applying ...
58 views

### Probability that a jar with 200 jelly beans with 40 colors has at least one of every color.

I am trying to calculate the probability that a jar with 200 jelly beans and 40 possible jelly bean colors does not feature all colors at least 1 time. I am aware of the way to solve this using ...
45 views

### Counting the amount of injective functions with restriction

First I will say that the exact same question has been uploaded to the site already few years ago but it seems no final answer were given there. Let $$X = \{a,b,c\}, \quad Y = \{1,2,3,4,5,6,7\}.$$ I ...
• 21