1. What Is a Combination?
A combination is a selection of objects where the order does not matter. We only care about which objects are chosen, not the arrangement.
Combinations are useful when forming teams, choosing items, or selecting members from a group.
1.1. Understanding 'Order Does Not Matter'
If two selections contain the same objects but in different orders, they are counted as the same combination.
Example: Picking A, B is the same as picking B, A. So both count as one combination.
1.2. Simple Real-Life Examples
Example: Choosing 2 friends to go to a competition. Selecting Riya and Arjun is the same as selecting Arjun and Riya.
Example: Choosing 3 ice-cream flavours out of 5. The order in which we choose them does not make a difference.
2. Formula for nCr
The number of ways to choose \(r\) objects from \(n\) objects without considering order is given by the combination formula.
2.1. Definition and General Formula
The formula for combinations is:
\( ^nC_r = \frac{n!}{r!(n-r)!} \)
This formula comes from dividing permutations by \(r!\) to remove repeated arrangements.
2.2. Special Values: nC0, nCn, nC1, nC2
Some useful values are:
- \( ^nC_0 = 1 \) (choosing nothing)
- \( ^nC_n = 1 \) (choosing all)
- \( ^nC_1 = n \)
- \( ^nC_2 = \frac{n(n-1)}{2} \)
3. Properties of Combinations
Combinations have several simple properties that help in solving problems quickly.
3.1. Symmetry Property: nCr = nC(n−r)
The number of ways to choose \(r\) objects is the same as choosing the \((n - r)\) objects to leave out.
Example: Choosing 3 students from 10 is the same as choosing the 7 students not selected.
3.2. Pascal’s Identity
Pascal’s identity states:
\( ^nC_r = ^{n-1}C_{r-1} + ^{n-1}C_r \)
This identity is used in Pascal’s triangle.
3.3. Sum of Combinations (Light Intuition)
When we add all combinations for \(r = 0\) to \(n\), we get:
\( ^nC_0 + ^nC_1 + ^nC_2 + \, \dots \, + ^nC_n = 2^n \)
This represents the total number of ways to choose any subset from \(n\) objects.
4. Relation Between Permutations and Combinations
Permutations count ordered arrangements. Combinations count unordered selections. They are connected through a simple formula.
4.1. Using nPr = nCr × r!
The relation between permutations and combinations is:
\( ^nP_r = ^nC_r \times r! \)
This means we first choose the objects, then arrange them.
4.2. Understanding 'Select First, Arrange Later'
To form a permutation of \(r\) items from \(n\):
- Select \(r\) items → \( ^nC_r \)
- Arrange those \(r\) items → \( r! \)
Multiplying gives the permutation formula.
5. Typical Combination Situations
Combinations appear in many selection-based problems. Here are some common situations.
5.1. Choosing Teams and Groups
Example: Choosing 4 players out of 10 for a team:
\( ^{10}C_4 \)
Since the order of selection does not matter, we use combinations.
5.2. Selecting Items Under Conditions
Example: Choosing 3 books from a shelf of 7, but at least one must be a novel. We break into cases and use combinations to count each case.