Quick Summary
- The binomial theorem expands (a + b)ⁿ without multiplying it out term by term.
- Each term has the form C(n,r) · aⁿ⁻ʳ · bʳ, where C(n,r) = n! / [(n−r)!r!].
- Pascal's Triangle gives you the same coefficients — useful for small n without a calculator.
- You'll usually only need the first few terms, not the whole expansion.
- To find one specific term (e.g. the coefficient of x³), pick out the right value of r using the powers.
- Watch signs carefully when b is negative — it flips every other term.
- You can be asked to work backwards: given a coefficient, find an unknown constant in the brackets.
1. What Is the Binomial Expansion?
"Binomial" just means two terms — like (a + b). If you wanted to expand
(a + b)² by hand, you'd get a² + 2ab + b². For (a + b)³, you'd get a³ + 3a²b + 3ab² + b³.
Notice a pattern? The powers of a count down, the powers of b count up,
and the numbers in front (the coefficients) follow a set pattern.
Now imagine doing this for (a + b)¹⁰ by hand — multiplying it out ten times would take forever
and you'd almost certainly make an arithmetic slip. The binomial theorem is the shortcut that
gives you every term directly, using a formula for the coefficients instead of grinding through
the multiplication.
Picture (a + b)ⁿ as choosing, n times, whether you "pick" an a or a "pick" a b, then multiplying your
choices together. C(n,r) simply counts how many different ways you could end up picking b
exactly r times (and a the other n−r times). That's why it's also called "n choose r" — you're literally
choosing r slots out of n to be the b's.
Notation you'll see
C(n,r) might be written a few different ways depending on your textbook or calculator —
they all mean exactly the same thing:
- ⁿCr
- ₙCr
- (n choose r), written as a stacked fraction-like bracket with n on top and r below
Practice Question 1
Write out the general term of the expansion of (a + b)⁷ that contains b⁴. Don't simplify the coefficient yet — just write the term using the formula.
2. Pascal's Triangle
Pascal's Triangle is a visual, no-formula way to get the same binomial coefficients. Each row
corresponds to a power n, and each number in that row corresponds to a C(n,r) value.
- The top row (just the number 1) is row 0 — it matches (a+b)⁰.
- Every row starts and ends with a 1.
- Every other number is the sum of the two numbers diagonally above it.
| 1 |
| |
| 1 | 1 |
| 1 | 2 | 1 |
| 1 | 3 | 3 | 1 |
| 1 | 4 | 6 | 4 | 1 |
| 1 | 5 | 10 | 10 | 5 | 1 |
Rows 0 to 5 — row 4 (1, 4, 6, 4, 1) gives the coefficients for (a+b)⁴
3. Expanding a Full Bracket, e.g. (3 − 2x)⁵
4. Finding the Coefficient of a Single Term
one specific term
How to choose the right r
where
- (a + bx)ⁿ
- (a + bx²)ⁿ
- (a + b/x)ⁿ
What to Memorise