Welcome to the foundational module on Combinations, a crucial topic in quantitative aptitude for competitive exams like the CAT. Understanding combinations allows us to count the number of ways to select items from a larger set where the order of selection does not matter. This skill is vital for solving probability, permutation, and various counting problems.

In mathematics, a combination is a selection of items from a set where the order of selection does not matter. For example, if you are choosing 2 fruits from a basket containing an apple, a banana, and a cherry, the combination {apple, banana} is the same as {banana, apple}. We are interested in the group of items, not the sequence in which they were picked.

The number of combinations of selecting 'r' items from a set of 'n' distinct items is denoted by C(n, r), nCr, or $\binom{n}{r}$. The formula is derived from permutations and is given by:

$C(n, r) = \frac{n!}{r!(n-r)!}$

Where '!' denotes the factorial (e.g., 5! = 5 × 4 × 3 × 2 × 1). It's important to remember that 0! is defined as 1.

Suppose you have a group of 5 friends, and you need to choose 3 of them to form a team for a quiz competition. The order in which you choose them doesn't matter; it's the same team regardless of selection order. Here, n = 5 (total friends) and r = 3 (friends to be chosen).

Using the combination formula:

$C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{(5 \times 4 \times 3 \times 2 \times 1)}{(3 \times 2 \times 1)(2 \times 1)} = \frac{120}{(6)(2)} = \frac{120}{12} = 10$

Therefore, there are 10 different ways to choose a team of 3 friends from a group of 5.

A common mistake is confusing combinations with permutations. Always ask yourself: 'Does the order of selection matter?' If not, use combinations. If yes, use permutations.

Practice simplifying factorials. Often, you can cancel out terms to make calculations easier. For instance, in C(10, 3), you can write 10! / (3! * 7!) as (10 * 9 * 8 * 7!) / (3 * 2 * 1 * 7!), canceling out 7!.