Intereting Posts

Size of a union of two sets
Generalized Logarithmic Integral – reference request
What does it means to multiply a permutation by a cycle? $\pi(x_1\cdots x_n)\pi^{-1}=(\pi(x_1)\cdots\pi(x_n))$
Is there an elementary way to see that there is only one complex manifold structure on $R^2$?
Evaluate these infinite products $\prod_{n\geq 2}(1-\frac{1}{n^3})$ and $\prod_{n\geq 1}(1+\frac{1}{n^3})$
How to show that the 3-cycles $(2n-1,2n,2n+1)$ generate the alternating group $A_{2n+1}$.
Homology – why is a cycle a boundary?
understanding the basic definition
An Inequality Involving Bell Numbers: $B_n^2 \leq B_{n-1}B_{n+1}$
e as sum of an infinite series
Eigenvalues of an $n\times n$ symmetric matrix
How to solve this operation research problem using dual simplex method?
Maximizing the number of points covered by a circular disk of fixed radius.
Problems with fake proofs of limit of sequences
Largest circle between $y=x^n$ and $y=\sqrt{x}$

You are sorting assigning 6 people, A, B, C, D, E and F, into 3 different hotel rooms. How many ways can they be sorted such that A is in the same room with C, and B is not in the same room with D? (Some hotel rooms may be empty.)

7C5 * 2! * 5! – 6C4 * 2! * 2! *4!

=5040-1440

- How many ways to divide group of 12 people into 2 groups of 3 people and 3 groups of 2 people?
- There are apparently $3072$ ways to draw this flower. But why?
- Relation between exclusive-OR and modular addition in a specific function
- Finding Expressively Adequate truth Functions
- Counting the exact number of coin tosses
- Permutation with Duplicates

=3600

- Exponential Generating Function of the numbers $r(n)$
- The probability that each delegate sits next to at least one delegate from another country
- $p$ divides $ax+by+cz$
- Traveling salesman problem: a worst case scenario
- Subset of natural numbers such that any natural number except 1 can be expressed as sum of two elements
- Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
- A combinatorial inequality
- How many ways are there to shake hands?
- Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$
- Minimum number of points chosen from an $N$ by $N$ grid to guarantee a rectangle?

Another solution.

Since A and C are in same room we have to assign 5 to 3 rooms.

It can be done in $3^5$ ways.This also contains cases when B and D are in same room.

The count of ways is $3^4$ because we have to assign 4 to 3 room.

Subtracting we get $3^5 – 3^4 = 162$

There are $3$ ways to assign B to a room, and then $2$ ways to assign D. That leaves the pair AC and the individuals E and F to be assigned; each of these three ‘people’ can be assigned to any of the three rooms, so they can be assigned in $3^3$ ways. The total number of allowable assignments is therefore $$3\cdot2\cdot3^3=2\cdot3^4=162\;.$$

Note that your answer can’t possibly be right: if there were no restrictions, each of the $6$ people could be assigned to any of the $3$ rooms; that’s a $3$-way choice made $6$ times, so it can be done in $3^6=729$ ways. Thus, your answer is way bigger than the number of possible assignments when there are no restrictions at all. The number of assignments satisfying the restrictions on A, B, C, and D must be smaller than $729$.

- Solve Burgers' Equation with side condition.
- Is the equality $1^2+\cdots + 24^2 = 70^2$ just a coincidence?
- On the decomposition of stochastic matrices as convex combinations of zero-one matrices
- Context for Russell's Infinite Sock Pair Example
- The Radon-Nikodym derivative of a measure such that $|\int f'\,d\mu|\le \|f\|_{L^2}$ for $f\in C^1$
- Rational Roots of Riemann's $\zeta$ Function
- If $f$ is continuous and injective on an interval, then it is strictly monotonic- what's wrong with this proof?
- Evaluating $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$
- Countable set having uncountably many infinite subsets
- Other Algebraically Independent Transcendentals
- Rigorous Text in Multivariable Calculus and Linear Algebra
- Is this GCD statement true?
- If the tensor product of two modules is free of finite rank, then the modules are finitely generated and projective
- If every convergent subsequence converges to $a$, then so does the original bounded sequence (Abbott p 58 q2.5.4 and q2.5.3b)
- Minimizing the variance of weighted sum of two random variables with respect to the weights