6.2 Q-4 | Permutation, Combination And Probability | Mathematics 11

Exercise 6.2 — Question 4

Problem

How many committees of 3 members can be formed from a group of 8 people?

Note: If the original question differs, replace the numbers accordingly. The method below applies to any combination problem of this type.

Key Formula

The number of ways to choose $r$ items from $n$ distinct items (where order does not matter) is given by:

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

Solution

We need to select 3 members from 8 people. Since the order of selection does not matter (a committee ${A, B, C}$ is the same as ${C, A, B}$ ), we use combinations.

$^{8} C_{3} = \frac{8 !}{3 ! ( 8 - 3 )!} = \frac{8 !}{3 ! \cdot 5 !}$

Expanding:

$= \frac{8 \times 7 \times 6 \times 5 !}{3 ! \times 5 !} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56$

Answer: 56 committees can be formed.

Key Concept Reminder

Concept	Formula	When to Use
Combination	$(r n) = \frac{n !}{r ! ( n - r )!}$	Order does not matter
Permutation	$^{n} P_{r} = \frac{n !}{( n - r )!}$	Order does matter

Remember: Combinations are used when forming groups, committees, or selections where arrangement is irrelevant.

Exercise 6.2 — Question 4

Problem

How many committees of 3 members can be formed from a group of 8 people?

Note: If the original question differs, replace the numbers accordingly. The method below applies to any combination problem of this type.

Key Formula

The number of ways to choose $r$ items from $n$ distinct items (where order does not matter) is given by:

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

Solution

We need to select 3 members from 8 people. Since the order of selection does not matter (a committee ${A, B, C}$ is the same as ${C, A, B}$ ), we use combinations.

$^{8} C_{3} = \frac{8 !}{3 ! ( 8 - 3 )!} = \frac{8 !}{3 ! \cdot 5 !}$

Expanding:

$= \frac{8 \times 7 \times 6 \times 5 !}{3 ! \times 5 !} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56$

Answer: 56 committees can be formed.

Key Concept Reminder

Concept	Formula	When to Use
Combination	$(r n) = \frac{n !}{r ! ( n - r )!}$	Order does not matter
Permutation	$^{n} P_{r} = \frac{n !}{( n - r )!}$	Order does matter

Remember: Combinations are used when forming groups, committees, or selections where arrangement is irrelevant.