How Many Ways to Fit Six People in Four Rooms?

  • MHB
  • Thread starter anemone
  • Start date
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    2016
In summary, to comfortably accommodate six people in four rooms, it is important to maximize space and plan carefully. This can be done by using bunk beds or trundle beds, utilizing common areas as additional sleeping space, and setting ground rules for privacy. Tips for maximizing space in a small living area include using multi-functional furniture, utilizing vertical space, and keeping clutter to a minimum. The most common room configurations include a combination of two bedrooms and a pull-out sofa or air mattress, or three bedrooms with two people in each room. Safety considerations for shared living spaces include setting up fire escape routes, ensuring functioning locks, and being mindful of allergies and medical conditions.
  • #1
anemone
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MHB
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Here is this week's POTW:

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In how many different ways can six people be accommodated in the four rooms, if each room can accommodate a maximum of four people?

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
Congratulations to kaliprasad for his correct solution, which you can find as follows::)

Six persons can be accommodated in 4 rooms in $4^6=4096$ ways without any constraints.

We need to rule out the number of ways 6 or 5 persons can be accommodated in a room.

Number of ways 6 persons can be accommodated in one room = 4 as in 4 ways we can choose the room for all persons.

If one room has 5 persons, and the other one room has one person. Then the 5 persons can be chosen in 6 ways, and the room for 5 persons can be chosen in 4 ways and then room for one person can be chosen in 3 ways. So the number of ways for such cases $= 6 \times 4 \times 3 = 72$.

Therefore the number of ways atmost 4 persons can be in a room = 4096- 4 - 72 = 4020.
 

Related to How Many Ways to Fit Six People in Four Rooms?

1. How can six people comfortably fit in only four rooms?

The key to accommodating six people in four rooms is efficient use of space and careful planning. One option is to use bunk beds or trundle beds to maximize sleeping space in the bedrooms. Another option is to utilize common areas, such as a living room or den, as additional sleeping space by setting up air mattresses or pull-out sofas.

2. What are some tips for maximizing space in a small living area?

Some tips for maximizing space in a small living area include using multi-functional furniture, such as a coffee table with storage or a sofa bed, utilizing vertical space with shelves or wall-mounted storage, and keeping clutter to a minimum to create a more open and spacious feel.

3. Are there any specific room configurations that work best for accommodating six people?

The most common room configurations for accommodating six people are a combination of two double or queen-sized bedrooms and a pull-out sofa or air mattress in a living area. Another option is to have three bedrooms with two people in each room. Ultimately, the best configuration will depend on the layout and size of the rooms.

4. How can I ensure privacy for all six people in a shared living space?

If privacy is a concern, it is important to establish ground rules and boundaries before the trip. This may include setting designated quiet times for sleeping, establishing a schedule for bathroom use, and respecting each other's personal space. Additionally, using room dividers or curtains can help create a sense of privacy in a shared living space.

5. Are there any safety considerations for accommodating six people in four rooms?

When sharing a living space with multiple people, it is important to be mindful of potential safety concerns. This may include setting up designated fire escape routes, ensuring that all doors and windows have functioning locks, and discussing any allergies or medical conditions that may require special accommodations. It is also a good idea to have a first aid kit and emergency contact information readily available in case of any accidents or emergencies.

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