# Combinatorics: Order in a line by 2 conditions

• MHB
• Lancelot1
In summary, the conversation discussed the different shapes of pasta and their weights in a box. There was a question about the probability of a random ordering of the packages on a shelf where pasta packages of the same weight would be next to each other, and/or each shape of pasta would be separate. The exact meaning of "and/or" was unclear, but it was interpreted as either Situation 1 (packages of the same weight next to each other) or Situation 2 (each shape of pasta separate), or both together. The conversation also mentioned that this is a probability question with a combinatorial problem and provided the number of possibilities for both situations. The issue with the weights was discussed, and it was suggested to treat them as groups and
Lancelot1
Hi all,

I need some help with this one:There are 3 shapes of pasta: 1,2,3.

In a box there are 3 packages of pasta of shape 1, with different weights: 300 gr, 400 gr, 500gr.
In addition, there are 5 packages of paste of shape 2, with weights: 300gr, 350gr, 400gr, 500gr, 600gr,
and 4 packages of pasta of shape 3, with weights 300gr, 350gr, 400gr, 500gr.

What is the probability that a random ordering of the packages on a shelf will be such that pasta packages of the same weight will be one next to another, and/or each shape of pasta will be separate ?

The and/or part of the question is unclear to me. My interpretation is to count all the possibilities in which the same weight is one next to another, or the shape is separate or both. Does it makes sense ?

This is a probability question, but the main problem is combinatorical.

The number of possibilities is clearly :

$(3+5+4)! = 12!$

The number of possibilities for separate shapes is:

$3!\cdot 3!\cdot 4!\cdot5!$

i.e, 6 possibilities to order the shapes, with all the inner possibilities within each shape, right ?

My problem is with the weights...

The separate shape rationale looks good to me. I find the and/or language confusing, as these are usually distinct calculations. My instinct is to treat these as two problems since I can't think of a way to combine these logically and I've never seen a situation where this happens. Maybe it means this.

Situation 1: Random ordering of the packages on a shelf will be such that pasta packages of the same weight will be one next to another
Situation 2: Each shape of pasta will be separate

"and/or" could mean either Situation 1, Situation 2, or both Situation 1 and Situation 2 together. This is treating it as a union.

As for the weights, if we think of each weight as a group then there are 3 300gr packages, 2 350gr, etc. I think you can apply the same approach to this kind of grouping as you did to the package groupings.

## 1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and arranging objects or events in a systematic way.

## 2. What is meant by "order in a line by 2 conditions" in combinatorics?

This refers to the process of arranging a set of objects or events in a line, while also taking into account two specific conditions or restrictions.

## 3. What are some common examples of combinatorics problems involving order in a line by 2 conditions?

Some common examples include arranging a group of people in a line based on their height and age, arranging a set of letters in a specific word while also considering the alphabetical order, and arranging a deck of cards in a specific order while also following the suit and rank rules.

## 4. How is combinatorics used in real life?

Combinatorics has numerous applications in various fields such as computer science, statistics, economics, and genetics. It is used to solve problems related to scheduling, optimization, data analysis, and more.

## 5. What are some strategies for solving combinatorics problems involving order in a line by 2 conditions?

Some common strategies include creating a visual representation, using a systematic approach such as a tree diagram or a table, and breaking down the problem into smaller, more manageable parts. It is also helpful to identify any patterns or relationships between the objects or events being arranged.

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