Combining combinations

Understanding Combinations

Combinations refer to the selection of items without considering the order in which they are selected.
Combinations apply to various areas of maths such as probability, statistics, and combinatorics.
Notation for combinations is often represented as nCr, which gives the number of ways ‘r’ items can be selected from ‘n’ items.

Attributes of Combinations

Combinations only account for unique selections; order does not matter.
A key formula for combinations is given by nCr = n! / [(n-r)! * r!], where n is the total number of items, r is the number of items to select, and ‘!’ represents the factorial operator.
The number of combinations is always less than or equal to the number of permutations (where order matters).

Combining Combinations and the Binomial Theorem

Combinations are heavily used in the expansion of binomial expressions through the binomial theorem.
Each term in the expansion corresponds to a particular combination, resulting in binomial coefficients.
The binomial theorem helps to expand expressions of the form (a+b)^n efficiently, and these coefficients can be found using Pascal’s triangle or by using the combination formula cited above.

Practical Application of Combinations

Combinations are used widely in mathematical problems dealing with selection or arrangement.
Problems involving combinations often involve deciding how many ways something can occur, given a certain number of elements and spaces.
Solving problems may involve the use of the combination formula, but it’s equally important to correctly identify when to use combinations - particularly as opposed to permutations.

Problems Involving Combination of Combinations

Sometimes problems may involve combining different combinations - essentially calculating the combinations of two or more different groups.
For instance, problems could involve selecting items from different groups where the order of selection doesn’t matter, thus calling for the combination of combinations.
In these cases, individual combinations are calculated and then added or multiplied depending on whether the selections are dependent or independent. Understanding the problem scenario is critical here.

Practising Combinations

Mastery of combinations and their application will come with practising problems.
Use textbooks, online resources, and past papers to practice different problems and hone your understanding.
Remember to review and understand the principle behind the formula; don’t just memorise it. Combinations and their applications are fundamental building blocks for deeper mathematical concepts.