Find the number of seven-letter words that use letters from the set $\{\alpha,\beta,\gamma,\delta, \epsilon\}$ and contain at least one each of $\alpha$, $\beta$, and $\gamma$. My attempt: using inclusion/exclusion Let A denotes $\alpha$ is missing, B denotes $\beta$ is missing, and C denotes $\gamma$ is missing. Then, $$\begin{aligned}|A\cup B\cup C|&=|A|+|B|+|C|-(|AB|+|AC|+|BC|)+|ABC|\\ &= 2^7+2^7+2^7-(1+1+1)-0\\ &= 381\end{aligned}$$ $|U|= […]

Given a multiset $S$ of $O$ ones and $Z$ zeros, I’d like to count the number of permutations of $S$ that when partitioned into length $T$ segments have at least one segment that is all ones. ($T$ must of course divide $O+Z$). For example, given the multiset $\{1,1,1,1,1,0,0,0,0,0,0,0,0,0,0\}$ with 5 ones and 10 zeros, the […]

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 + […]

Find the number of distributions of five red balls and five blues balls into 3 distinct boxes with no empty boxes allowed so I know if I have just 5 identical balls and 3 distinct boxes, the answer would be $\binom{7}{2}$ but because i have another set of 5 balls, I’m unsure how to proceed […]

So to solve this, I started with the total and subtract where some digit occurs only once or twice. Total = $5^5$ Digits are only used once $= 5! = 120$ Digits occur twice: I’m starting to think this is where the inclusion exclusion comes into play because as lulu pointed out, AABCD can occur […]

This is one of a set of several problems in my book I am having difficulty not just solving, but also understanding the provided solutions. The given answer is $462 – 336 + 15 = 141.$ I’ll try and see where the terms of the equality above come from. $[6] = \{1, 2, 3, 4, […]

I have been struggeling for some time with the following exercise from a book in discrete mathematics. It is an inclusion-exclusion principle exercise. How many permutations of 1, 1, 2, 2, 3, 3, 4, 4, 5, 5 are there in which no two adjacent numbers are equal? I have come thus far: Let S be […]

Given a binary string consisting of $O$ ones and $Z$ zeros, I’d like to count the number of permutations of that string where all of the ones are within a window (consecutive positions) of length $L$. For example, with 5 ones and 6 zeros, there are 81 permutations where the ones are all within $l=7$ […]

I would like to show that for any positive integers $d_1, \dots, d_r$ one has $$ \sum_{i=1}^r (-1)^{i+1}\biggl( \sum_{1\leq k_1 < \dots < k_i \leq r} \gcd(d_{k_1}, \dots , d_{k_i})\biggr) ~\leq~ \prod_{i=1}^r\biggl( \prod_{1\leq k_1 < \dots < k_i \leq r} \gcd(d_{k_1}, \dots , d_{k_i}) \biggl)^{(-1)^{i+1}}. $$ Note that the rhs of the upper inequality is […]

I need to find number of permutations $p$ of set $\lbrace 1,2,3, \ldots, n \rbrace$ such for all $i$ $p_{i+1} \neq p_i + 1$. I think that inclusion-exclusion principle would be useful. Let $A_k$ be set of all permutation that for every permutation $a$ in this set $a_{k+1} \neq a_k + 1$. So our answer […]

Intereting Posts

Minimal polynomial of restriction to invariant subspace divides minimal polynomial
Big Greeks and commutation
The product of two numbers that can be written as the sum of two squares
The primes $s$ of the form $6m+1$ and the greatest common divisor of $2s(s-1)$
Are $4ab\pm 1 $ and $(4a^2\pm 1)^2$ coprime?
How to express $z^8 − 1$ as the product of two linear factors and three quadratic factors
Transpose map in $M(2,\mathbb{R})$
Prove that B is a basis for a topology
If N and every subgroup of N is normal in G then G/N is abelian .
Plane intersecting line segment
How to sort vertices of a polygon in counter clockwise order?
Dimension of the sum of subspaces
Is it possible to draw this picture without lifting the pen?
Proving $\sum_{k=1}^n\frac1{\sqrt k}<2\sqrt n$ by induction
Does the nine point circle generalise to some theorem about n-spheres and n-simplices?