1. PROGRAMME F7
BINOMIALS

2. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

3. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

4. Factorials and combinations
Three properties of combinatorial coefficients

5. Factorials and combinations
If n is a natural number then the product of the successive natural numbers:
is called n-factorial and is denoted by the symbol n!
In addition 0-factorial, 0!, is defined to be equal to 1. That is, 0! = 1

6. Factorials and combinations
There are different ways of arranging r different items in n different locations.
If the items are identical there are r! different ways of placing the identical items within one arrangement without making a new arrangement.
So, there are different ways of arranging r identical items in n different locations.
This denoted by the combinatorial coefficient

7. Factorials and combinations
Three properties of combinatorial coefficients

8. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

9. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

10. Binomial expansions
Pascal’s triangle
The general term of the binomial expansion

11. Binomial expansions
Pascal’s triangle
The following triangular array of combinatorial coefficients can be constructed where the superscript to the left of each coefficient indicates the row number and the subscript to the right indicates the column number:

12. Binomial expansions
Pascal’s triangle
Evaluating the combinatorial coefficients gives a triangular array of numbers that is called Pascal’s triangle:

13. Binomial expansions
A binomial is a pair of numbers raised to a power. For natural number powers these can be expanded to give the appropriate Binomial expansions:

14. Binomial expansions
Notice that the coefficients in the expansions are the same as the numbers in Pascal’s triangle:

15. Binomial expansions
The power 4 expansion can be written as:
or as:

16. Binomial expansions
The general power n expansion can be written as:
This can be simplified to:

17. Binomial expansions
The general term of the binomial expansion
The (r + 1)th term in the expansion of is given as:

18. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

19. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

20. The sigma notation
General terms
The sum of the first n natural numbers
Rules for manipulating sums

21. If a sequence of terms are added together:
The sigma notation
General terms
If a sequence of terms are added together:
f(1) + f(2) + f(3) f(r) f(n)
their sum can be written in a more convenient form using the sigma notation:
The sum of terms of the form f(r) where r ranges in value from 1 to n.
f(r) is referred to as a general term.

22. The sigma notation
General terms
The sigma notation form of the binomial expansion is.

23. The sigma notation
The sum of the first n natural numbers
The sum of the first n natural numbers can be written as:

24. The sigma notation
Rules for manipulating sums
Rule 1: Constants can be factored out of the sum
Rule 2: The sum of sums

25. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

26. Factorials and combinations
Binomial expansions
The sigma notation
The exponential number e

27. The exponential number e
The binomial expansion of is given as:

28. The exponential number e
The larger n becomes the smaller becomes – the closer its value becomes to 0.
This fact is written as the limit of as is 0.
Or, symbolically

29. The exponential number e
Applying this to the binomial expansion of gives:

30. The exponential number e
It can be shown that is a finite number whose decimal form is:
This number, the exponential number, is denoted by e.

31. The exponential number e
It will be shown in Part II that there is a similar expansion for the exponential number raised to a variable power x, namely:

32. Learning outcomes
Define n! and recognise that there are n! different combinations of n different items
Evaluate n! using a calculator and manipulate expressions involving factorials
Recognize that there are different combinations of r identical items in n locations
Recognize simple properties of combinatorial coefficients
Construct Pascal’s triangle
Write down the binomial expansion for natural number powers
Obtain specific terms in the binomial expansion using the general term
Use the sigma notation
Recognize and reproduce the expansion for ex where e is the exponential number