Permutations with restrictions : items must not be together

Basics of Permutations

A permutation refers to an arrangement of objects in a particular order.
The number of permutations is calculated by multiplying the number of choices available at each decision point.
When all the items are unique, the number of permutations is simply the factorial of the total items.

Permutation Restrictions: Items Must Not be Together

Sometimes, restrictions are set for permutations, such as certain items not being allowed to be together.
This type of problem is commonly solved in two steps: calculating the total permutations without restrictions and then subtracting the unwanted permutations.

Calculating Total Permutations

First, compute the number of total permutations without any restriction.
If you have n distinct items, the total number of permutations is n! (n factorial).

Calculating Unwanted Permutations

Treat the items that are not allowed to be together as a single item, and then calculate the permutations.
For instance, if there are m items out of n which are not supposed to be together, treat them as one single entity. Now, you have n-m+1 items in total.
The number of permutations for these (n-m+1) items is (n-m+1)!.

Removing Unwanted Permutations

Remember, the items we considered as one single entity can also arrange among themselves. There are m! ways to arrange these.
Therefore, unwanted permutations will be (n-m+1)! x m!.

Final Answer

To find the number of permutations where certain items are not together, subtract the unwanted permutations from the total permutations.
The required number of arrangements is thus n! - (n-m+1)! x m!.

Tips and Tricks

When calculating permutations where items should not be together, always first consider the total permutations without restrictions.
To take into account the restriction, treat the forbidden items as one item then subtract the resultant arrangements from the total.
Note that subtraction is critical here because it helps remove any permutations that violate the restrictions.
Regular practice with different scenarios and an understanding of combinatorial principles is key to mastering this concept.