Q.Verify that $^{10}C_5 + {}^{10}C_6 = {}^{11}C_6$

Combination: Choosing Without Ordering

Imagine you have three friends: A, B, and C. You need to pick two of them to form a committee. How many different committees can you make?

You could pick A and B, or A and C, or B and C. That's three committees. Notice something crucial: the committee {A, B} is exactly the same as {B, A}. Who gets picked first doesn't matter — only who is on the committee matters. This is the core idea of a combination: a selection where the order of selection does not matter.

The Intuition: "How many groups?"

A combination answers the question: "From a set of $n$ distinct items, how many ways can I choose $r$ of them, ignoring the order in which I pick them?"

Think of it as forming a team, a committee, a hand of cards, or a set of ingredients for a recipe. In all these cases, swapping two members around doesn't give you a new team — it's still the same group.

The Precise Statement

A combination is a selection of $r$ objects from a set of $n$ distinct objects, where the order of selection does not matter. The number of such combinations is denoted by:

and is given by the formula:

where $n!$ (read "n factorial") means $n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$, and $0! = 1$ by definition.

Why the $r!$ in the denominator?

This is the key to understanding combinations. If order did matter, the number of ways to pick $r$ objects from $n$ would be $n \times (n-1) \times \cdots \times (n-r+1)$, which is written as $\frac{n!}{(n-r)!}$. That's called a permutation.

But in a combination, every group of $r$ objects can be rearranged in $r!$ different orders. Since we don't care about order, we divide by $r!$ to "collapse" all those orderings into one single group.

So:

A Quick Example

From a class of 10 students, how many ways can you choose a team of 3?

Here $n=10$, $r=3$.

So there are 120 different teams possible.

A Common Mistake

The Symmetry Property

Notice that $\binom{n}{r} = \binom{n}{n-r}$. Choosing which 3 students to include is the same as choosing which 7 students to exclude. This symmetry often simplifies calculations: $\binom{10}{7} = \binom{10}{3} = 120$.

Summary

A combination counts groups, not sequences. The formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ is your tool for counting selections where order is irrelevant. Always ask yourself first: "Does swapping the items give a different outcome?" If no, you're dealing with a combination.