Permutations, Combinations and the Binomial Theorem

Counting arrangements when order matters and when it does not, and the binomial expansion — which turns out to be the same counting problem written as algebra.

หลักสูตรแกนกลาง / สสวท.: สถิติและความน่าจะเป็น (Statistics and Probability)ตรงกับ ม.4 หลักการนับเบื้องต้น (เพิ่มเติม) — 0606 รวมทฤษฎีบททวินามไว้ในเรื่องเดียวกัน
Common Core (US): S-CP, A-APR
Cambridge: IGCSE 0606 — Topics 11 and 12: Permutations and combinations; Series

9.1 When order matters, and when it does not

The two counts

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

A permutation is an arrangement, a combination is a selection. The only difference between the formulas is the $r!$ , which divides out the orderings you have decided not to count.

●
How to tell which one a question wants
- arrange, order, line up, code, password — a permutation
- choose, select, committee, team, hand of cards — a combination
- if swapping two of the chosen things gives a genuinely different outcome, order matters

ตัวอย่าง 1

From a class of $12$ , a committee of $4$ is chosen and then a president and a secretary are chosen from within the committee. In how many ways can this be done?

วิธีทำ

1.Choosing the committee ignores order: $^{12} C_{4} = 495$
2.Choosing two officers from four, in order: $^{4} P_{2} = 12$
3.Multiply, because the choices are made one after the other

ตอบ $495 \times 12 = 5940$

ตัวอย่าง 2

How many four-digit numbers greater than $3000$ can be formed from the digits $1, 2, 3, 4, 5$ without repetition?

วิธีทำ

1.The first digit must be $3$ , $4$ or $5$ — three choices
2.The remaining three places take any three of the four digits left: $^{4} P_{3} = 24$

ตอบ $3 \times 24 = 72$

เคล็ดลับ

Deal with the restriction FIRST. Filling the restricted place before the free ones turns almost every counting question into a short multiplication, while starting from the free places usually forces you to subtract cases afterwards.

📖 2 more sections of this chapter are being checked

The whole chapter is written; we read it back section by section before releasing it, because maths nobody has checked should not be on a public page. What is open is complete — nothing here is abridged.

Read it — now practise it

Generate a worksheet on this topic, or sit a timed mock paper.

Build a worksheet Practice arena