Exercise 6.2 — Question 1: Combinations

Key Formula

The number of ways to choose $r$ objects from $n$ distinct objects without regard to order is given by the combination formula:

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

where $0 \leq r \leq n$ .

Important Properties

Property	Formula
Symmetry	$^{n} C_{r} =^{n} C_{n - r}$
Choosing none	$^{n} C_{0} = 1$
Choosing all	$^{n} C_{n} = 1$
Choosing one	$^{n} C_{1} = n$
Pascal's identity	$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

Worked Examples

Example 1

In how many ways can a committee of 3 be selected from 7 people?

$^{7} C_{3} = \frac{7 !}{3 ! 4 !} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

Example 2

Evaluate $^{8} C_{5}$ .

$^{8} C_{5} =^{8} C_{3} = \frac{8 !}{3 ! 5 !} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

Example 3

In how many ways can 4 books be chosen from a shelf of 9 books?

$^{9} C_{4} = \frac{9 !}{4 ! 5 !} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$

Difference: Permutation vs Combination

	Permutation	Combination
Order matters?	Yes	No
Formula	$\frac{n !}{( n - r )!}$	$\frac{n !}{r ! ( n - r )!}$
Example	Arranging 3 from 5	Choosing 3 from 5

Exercise 6.2 — Question 1: Combinations

Key Formula

The number of ways to choose $r$ objects from $n$ distinct objects without regard to order is given by the combination formula:

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

where $0 \leq r \leq n$ .

Important Properties

Property	Formula
Symmetry	$^{n} C_{r} =^{n} C_{n - r}$
Choosing none	$^{n} C_{0} = 1$
Choosing all	$^{n} C_{n} = 1$
Choosing one	$^{n} C_{1} = n$
Pascal's identity	$^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$

Worked Examples

Example 1

In how many ways can a committee of 3 be selected from 7 people?

$^{7} C_{3} = \frac{7 !}{3 ! 4 !} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$

Example 2

Evaluate $^{8} C_{5}$ .

$^{8} C_{5} =^{8} C_{3} = \frac{8 !}{3 ! 5 !} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

Example 3

In how many ways can 4 books be chosen from a shelf of 9 books?

$^{9} C_{4} = \frac{9 !}{4 ! 5 !} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$

Difference: Permutation vs Combination

	Permutation	Combination
Order matters?	Yes	No
Formula	$\frac{n !}{( n - r )!}$	$\frac{n !}{r ! ( n - r )!}$
Example	Arranging 3 from 5	Choosing 3 from 5

6.2 Q-1

Exercise 6.2 — Question 1: Combinations

Key Formula

Important Properties

Worked Examples

Example 1

Example 2

Example 3

Difference: Permutation vs Combination

6.2 Q-1

Exercise 6.2 — Question 1: Combinations

Key Formula

Important Properties

Worked Examples

Example 1

Example 2

Example 3

Difference: Permutation vs Combination