Permutation, Combination and Probability Basics

Why This Chapter Matters

This chapter feels abstract only until you ask one question first: does order matter? That decision usually chooses the right method.

Permutation counts ordered arrangements. Combination counts unordered selections.
Use $n!$ for factorial-based counting, where $n! = n (n - 1) (n - 2) \dots 1$ .
Probability $=\frac{\text{favourable outcomes}}{\text{total outcomes}$ .
In digit-formation questions, the first place often has an extra restriction because zero may not be allowed there.
For at least one success, it is often faster to use the complement method.
When repeated items exist, divide by repeated factorial counts to remove overcounting.

Decide whether arrangement order matters.
Count the total outcomes first, then the favourable outcomes.
If there is a forbidden position or repeated object, handle that restriction explicitly.
Watch for replacement or no-replacement conditions.

Prompt: How many ways can $3$ people be arranged in a row?

Approach: The arrangement count is $3! = 6$ .

Prompt: How many ways can $2$ books be chosen from $5$ books?

Approach: Since order does not matter, use $_{5} C_{2} = 10$ .

Prompt: How many different $3$ -digit numbers can be formed using the digits $1, 2, 3, 4$ without repetition?

Approach: This is an arrangement of $3$ places from $4$ distinct digits, so the count is $_{4} P_{3} = 24$ .

Prompt: Two fair dice are thrown. What is the probability that the sum is $8$ ?

Approach: Favourable outcomes are $(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)$ , so probability $= \frac{5}{36}$ .

Ask whether order matters, then count carefully and simplify.