Permutations with restrictions: letters / items together

Basic Principles

A permutation is the arrangement of elements or objects in a particular order.
The process of creating permutations with certain elements or objects grouped together is often referred to as permutations with restrictions.
When restricting permutations to group specific items together, they are viewed as a single object during the arrangement process.

Grouping Letters/Items Together

Certain permutation problems may require letters or items to be grouped together. This means that these specific items always appear next to each other in the permutation.
For such problems, consider the group of items as a single entity or unit at first. The rearrangement then involves this unit and the remaining items.

Calculating Permutations

When we group n items together, we first calculate the permutations inside this group which is n! (n factorial), and then work out the permutations of the other items and the group (considered as a single item).
The total number of permutations is the product of these two numbers.
For instance, if we have 5 items and we need to group 2 of them together, we would have 2! arrangements of the group, and 4! arrangements of the total ‘items’ including the group. So, the total number of permutations is 2! × 4!.

Special Cases

If there are duplicate elements in the group or among the other items, those duplicates need to be accounted for as they reduce the total number of unique permutations.
To do this, we divide the total number of permutations by the factorials of the multiplicity of each recurring item or letter.

Examples and Applications

This concept is frequently applied in problems of arrangements such as seating arrangements, password settings, and various combinatorial problems in probability and statistics.
A strong grasp of permutations with restrictions can help solve complex problems in coding theory and cryptography too.

Revision Tips

Make sure to understand the principles deeply and apply in various types of problems to gain proficiency.
Regular practice using past papers and problem sets can greatly aid in mastering this topic.
It’s also helpful to check solutions with online calculators to ensure correctness during practice.
Don’t forget to watch tutorial videos if any concept is unclear. These visual aids can significantly help make complex topics easier to understand.