6.3 Combinations and Their Properties | Permutation, Combination And Probability | Mathematics 11

What is a Combination?

A combination is a selection of items from a set where order does not matter. The number of ways to choose $r$ items from $n$ distinct items is:

$^{n} C_{r} = (r n) = \frac{n !}{r ! ( n - r )!}$

where $0 \leq r \leq n$ .

Key Properties of Combinations

1. Symmetry Property

$^{n} C_{r} =^{n} C_{n - r}$

This means choosing $r$ items is equivalent to leaving out $n - r$ items.

Consequence: If $^{n} C_{p} =^{n} C_{q}$ , then either $p = q$ or $p + q = n$ .

Example: If $^{n} C_{15} =^{n} C_{7}$ , then $n = 15 + 7 = 22$ .

2. Pascal's Identity

$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

This identity shows how combinations in one row of Pascal's Triangle are built from the row above.

Example: $^{5} C_{2} +^{5} C_{3} =^{6} C_{3} = 20$ .

3. Special Values

$^{n} C_{0} =^{n} C_{n} = 1^{n} C_{1} = n$

Conditional Selection Problems

Fixed Item Included

If one specific item must be included in every selection of $r$ items from $n$ :

Fix that item (1 way)
Choose the remaining $r - 1$ items from the other $n - 1$ items

$Ways =^{n - 1} C_{r - 1}$

Complementary Counting

For conditions like "at least one", it is often easier to use:

$Favourable = Total - Complement$

Example: Committee of 5 from 6 men and 4 women with at least one man: $=^{10} C_{5} -^{4} C_{5} = 252 - 0 = 252$

(Since $^{4} C_{5} = 0$ , all committees automatically include at least one man here. For cases where the complement is non-zero, subtract the all-women count.)

Real-World Application: Diagonals of a Polygon

In an $n$ -sided polygon, any two vertices can be connected by a line segment. The total number of such segments is $^{n} C_{2}$ . Subtracting the $n$ sides gives the number of diagonals:

$Diagonals =^{n} C_{2} - n = \frac{n ( n - 1 )}{2} - n = \frac{n ( n - 3 )}{2}$

Example: A hexagon ( $n = 6$ ) has $\frac{6 \times 3}{2} = 9$ diagonals.

Real-World Applications

Context	How Combinations Apply
Lottery	Choosing 6 numbers from 49: $^{49} C_{6}$ possible tickets
DNA sequences	Selecting bases for a codon from available nucleotides
Cryptography	Choosing a subset of keys or characters for a cipher
Playlist selection	Choosing 5 songs from a library of 20: $^{20} C_{5}$ playlists

These applications all share the property that order does not matter — only which items are selected.

What is a Combination?

A combination is a selection of items from a set where order does not matter. The number of ways to choose $r$ items from $n$ distinct items is:

$^{n} C_{r} = (r n) = \frac{n !}{r ! ( n - r )!}$

where $0 \leq r \leq n$ .

Key Properties of Combinations

1. Symmetry Property

$^{n} C_{r} =^{n} C_{n - r}$

This means choosing $r$ items is equivalent to leaving out $n - r$ items.

Consequence: If $^{n} C_{p} =^{n} C_{q}$ , then either $p = q$ or $p + q = n$ .

Example: If $^{n} C_{15} =^{n} C_{7}$ , then $n = 15 + 7 = 22$ .

2. Pascal's Identity

$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

This identity shows how combinations in one row of Pascal's Triangle are built from the row above.

Example: $^{5} C_{2} +^{5} C_{3} =^{6} C_{3} = 20$ .

3. Special Values

$^{n} C_{0} =^{n} C_{n} = 1^{n} C_{1} = n$

Conditional Selection Problems

Fixed Item Included

If one specific item must be included in every selection of $r$ items from $n$ :

Fix that item (1 way)
Choose the remaining $r - 1$ items from the other $n - 1$ items

$Ways =^{n - 1} C_{r - 1}$

Complementary Counting

For conditions like "at least one", it is often easier to use:

$Favourable = Total - Complement$

Example: Committee of 5 from 6 men and 4 women with at least one man: $=^{10} C_{5} -^{4} C_{5} = 252 - 0 = 252$

(Since $^{4} C_{5} = 0$ , all committees automatically include at least one man here. For cases where the complement is non-zero, subtract the all-women count.)

Real-World Application: Diagonals of a Polygon

In an $n$ -sided polygon, any two vertices can be connected by a line segment. The total number of such segments is $^{n} C_{2}$ . Subtracting the $n$ sides gives the number of diagonals:

$Diagonals =^{n} C_{2} - n = \frac{n ( n - 1 )}{2} - n = \frac{n ( n - 3 )}{2}$

Example: A hexagon ( $n = 6$ ) has $\frac{6 \times 3}{2} = 9$ diagonals.

Real-World Applications

Context	How Combinations Apply
Lottery	Choosing 6 numbers from 49: $^{49} C_{6}$ possible tickets
DNA sequences	Selecting bases for a codon from available nucleotides
Cryptography	Choosing a subset of keys or characters for a cipher
Playlist selection	Choosing 5 songs from a library of 20: $^{20} C_{5}$ playlists

These applications all share the property that order does not matter — only which items are selected.