1. The Binomial Theorem

2. Expansion of Binomials

3. Binomial Expansion Coefficients
Pascal’s Triangle
Row

4. Binomial Coefficient

5. n-Factorial or n! n-Factorial For any positive integer n, and
Example Evaluate (a) 5! (b) 7!
Solution (a)
(b)

6. Simplifying r! 4 1

7. Simplifying

8. Simplifying 2 2

9. Binomial Coefficient The symbols and for the binomial
coefficients are read “n choose r”
The values of are the values in the nth
row of Pascal’s triangle.
So is the first number in the third row
and is the third.

10. Binomial Coefficient Binomial Coefficient
For non-negative integers n and r, with r < n,

11. Formulae
Factorial Formula
Multiplicative Formula

12. Evaluating Binomial Coefficients
Example Evaluate (a) (b)
Solution
(a)
(b)

13. The Binomial Theorem
Binomial Theorem
For any positive integers n,

14. Applying the Binomial Theorem
Example Write the binomial expansion of
Solution Use the binomial theorem

15. Applying the Binomial Theorem

16. Applying the Binomial Theorem
Example Expand
Solution Use the binomial theorem with
and n = 5,

17. Applying the Binomial Theorem
Solution

18. r th Term of a Binomial Expansion
rth Term of the Binomial Expansion
The rth term of the binomial expansion of (x + y)n,
where n > r – 1, is

19. Finding a Specific Term of a Binomial Expansion.
Example Find the fourth term of
Solution Using n = 10, r = 4, x = a, y = 2b in the
formula, we find the fourth term is

20. Pg 130 E.g 11(a)
Find, in ascending powers of x, the 1st 4 terms of the expansion (1+2x)n, where n > 2. Given that the coefficients of x3 and x2 are in the ratio 14 : 3, find n.
Using Multiplicative Formula:
Using Factorial Formula: 2 2

21. Pg 132 Q9(a)
In the expansion of (2+3x)n, the coefficients of x3 and x4 are in the ratio 8 : 15. Find n.
Using Factorial Formula: