8.2PureCore

Factorial Notation

Pascal’s triangle is slow for large powers. Factorials and the “choose” function $(r n)$ let you calculate any binomial coefficient directly — the engine behind the full binomial expansion.

25 min Video by Zeeshan Zamurred The Binomial Expansion

Edexcel AS level Maths: 8.2 Factorial NotationWatch the full walkthrough before the notes below.

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What you'll be able to do

Understand and evaluate factorials
Use the nCr (binomial coefficient) formula
Compute binomial coefficients efficiently
Connect nCr to rows of Pascal’s triangle

Factorials

The factorial $n!$ means the product of all positive integers up to $n$ . By convention $0! = 1$ .

n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1

For example

5! = 5 \times 4 \times 3 \times 2 \times 1 = 120

The choose function

The binomial coefficient $(r n)$ (read “ $n$ choose $r$ ”) counts how many ways to choose $r$ items from $n$ . It is exactly the coefficient of each term in a binomial expansion.

(r n) = \frac{n !}{r ! ( n - r )!}

On a calculator this is the

n C r

button.

\frac{5 !}{2 ! 3 !} = \frac{120}{2 \times 6} = \frac{120}{12}

Answer

10

Tip — Cancel before multiplying: 5!/(3!) = 5 × 4 leaves much smaller numbers than computing 120.

Link to Pascal’s triangle

The entries in row $n$ of Pascal’s triangle are exactly $(0 n), (1 n), \dots, (n n)$ . So the formula reproduces the triangle for any size, instantly.

(0 4) = 1

(1 4) = 4

(2 4) = 6

(3 4) = 4

(4 4) = 1

Answer

1, 4, 6, 4, 1

Formula recap

n! = n \times (n - 1) \times \dots \times 1

Factorial; 0! = 1.

(r n) = \frac{n !}{r ! ( n - r )!}

Binomial coefficient (nCr).

(r n) = (n - r n)

Symmetry of the coefficients.

Common mistakes to avoid

Thinking 0! = 0.

By definition 0! = 1.

Computing huge factorials in full before dividing.

Cancel the larger factorial first to keep the arithmetic small.

Key takeaways

n! is the product of all integers from n down to 1; 0! = 1.
nCr = n! / (r!(n − r)!) counts choices and gives binomial coefficients.
Row n of Pascal’s triangle is nC0, nC1, …, nCn.
Use the symmetry nCr = nC(n − r) to save work.

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