Binomial Expansions Lesson 1

Binomial Expansions Factorials n! is the symbol for the result when all the numbers between 1 and n are multiplied together.

n! = 1 × 2 × 3 × ...( n − 3)(n − 2)(n − 1)n where n∈N Special case: 1!= 0!= 1 Example:

4!= 4 × 3 × 2 × 1 4!= 24

Factorials can grow at an exponential rate. This means that they can get very large very fast.

E.g .

65!= 8.248 × 10 90

To overcome this problem we can simplify factorial expressions 10! 10 × 9 × 8 × 7 × ... × 1 Example : = 8! 8 × 7 × 6 × ... × 1 we can cancel out 8! 10 × 9 = 1 = 90

We can apply this simplification method in the permutation combination formulas when n is large. n

Pr =

n! (n − r )!

Evaluate Example:

Example:

n

and

100

Cr =

n! r!( n − r )!

100! 2!×98! 100 × 99 = 2 ×1 = 4950

C2 =

Write as a product 24!− 22! We note that 24!= 24 × 23 × 22 × 21 × ... × 1 = 24 × 23 × 22! Therefore 24!− 22!= 24 × 23 × 22!− 22! = 22!(24 × 23 − 1) = 22!× 551

Sigma: page 59, exercise 4.1