## Fundamental Principle

The product of first n positive integer, can be called as n factorial, and noted as n!

Note, the special case defined,

When we pick r objects from n objects, if the order matters, it is called as permutation, and we notate it as nPr.

If the order is unimportant, we can use combination, which is nCr.

The extra r! is the permutation in the specific set drawn out.

Note,

There will be n+1 th term.

The sum of power of x and y will equal to n.

The r+1 th term can be called as the general term, in the descending power of x.

The coefficients, r=0, 1 … are called binomial coefficients.

### E1 Example

Q

Expand

in the descending powers of x.

A

### E2 Example

Q

Expand

in the descending powers of x.

A

### E3 Example

Q

Expand

in the ascending powers of x.

A

## Pascal Triangle

0 | 1 |  |  |  |  |  |
1 | 1 | 1 |  |  |  |  |
2 | 1 | 2 | 1 |  |  |  |
3 | 1 | 3 | 3 | 1 |  |  |
4 | 1 | 4 | 6 | 4 | 1 |  |
5 | 1 | 5 | 10 | 10 | 5 | 1 |
6 | 1 | 6 | 15 | 20 | 15 | 6 | 1

Two adjacent cells sum will be the new value of the next row.

For example, in row of 4, the 2nd value is 1+4=5. 3rd value is 4+6=10.

### E4 Example

Q

In the expansion

, find (1) The coefficient of x3. (2) The constant term.

A

The general term will be:

(1)

Set

Hence, coefficient of x3 is 816,293,376.

(2)

Set

Hence, the constant term is -2,857,026,816.

### E5 Example

Q

Let n be a positive integer that in the expansion of

. The coefficient of the third term is 1,029.

Find the value of n and the coefficient of x12.

A

The 3rd term is 3 - 1 = 2,

Hence, the general term is

Set 21 - 3r = 12, r = 3

The coefficient of x12 is 12,005.

## Non positive power expansion

If n < 0 and |x| < 1, another formula can be used:

The negative or fractional n is a series that does not terminate, an infinite series.

The series is convergent, as the limit of it sums only when |x| < 1.

The expansion is not valid for

can be used instead.

### E6 Example

Q

Expand the followings in ascending power of x up to and including the term x3.

State the range of values of x for which the expansion is valid.

A

For

### E7 Example

Q

Expand the followings in ascending power of x up to and including the term x3.

State the range of values of x for which the expansion is valid.

A

For

### E8 Example

Q

Find the value of 4.0167 with estimation, by substituting 0.004 using the binomial expansion.

Correct the answer to the nearest 2 decimal place.

A

(correct to nearest 2 decimal places)