MHB Combinatorics: Order in a line by 2 conditions

[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...

 

[Jameson]

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.