Probability & Statistics 1 (S1)

Permutations and Combinations: Arrangements and Selections

This section covers the fundamental concepts of permutations and combinations, which are essential for calculating the number of possible outcomes in various scenarios. Understanding the difference between arrangements and selections is crucial.

1. Definitions

Permutation: An arrangement of objects in a specific order. The order of the objects matters.

Combination: A selection of objects where the order does not matter. The order of the selected objects is irrelevant.

2. Calculating Permutations

The number of permutations of n distinct objects taken r at a time is denoted by $P(n, r)$ or $_nP_r$ and is calculated as:

$$P(n, r) = \frac{n!}{(n-r)!}$$

where '!' denotes the factorial (e.g., 5! = 5 x 4 x 3 x 2 x 1).

Example: How many ways can you arrange 3 books on a shelf?

Here, n = 3 (total number of books) and r = 3 (number of books to arrange). So,

$P(3, 3) = \frac{3!}{(3-3)!} = \frac{3!}{0!} = \frac{6}{1} = 6$

The possible arrangements are: ABC, ACB, BAC, BCA, CAB, CBA.

3. Calculating Combinations

The number of combinations of n distinct objects taken r at a time is denoted by $C(n, r)$ or $_nC_r$ or $\binom{n}{r}$ and is calculated as:

$$C(n, r) = \frac{n!}{r!(n-r)!}$$

Example: How many ways can you choose 2 students from a group of 5?

Here, n = 5 (total number of students) and r = 2 (number of students to choose). So,

$C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{120}{2 \times 6} = \frac{120}{12} = 10$

The possible combinations are: AB, AC, AD, BC, BD, CD.

4. Key Differences between Permutations and Combinations

| Feature | Permutation | Combination |

Feature	Permutation	Combination
Order	Order matters	Order does not matter
Formula	$P(n, r) = \frac{n!}{(n-r)!}$	$C(n, r) = \frac{n!}{r!(n-r)!}$
Example	Arranging letters in a word	Selecting a committee

5. Applications

Permutations and combinations are used in various applications, including:

• Calculating the number of possible passwords.
• Determining the number of ways to select a team from a larger group.
• Calculating the probability of specific outcomes in games.
• Analyzing data and making inferences.