Chapter 1 Permutations and Combinations



Chapter 2 The Binomial Expansion

[2.1 The Binomial Theorem]

[[2.1.1 Binomial Expansion]]

(a + b)0 = = 1

(a + b)1 = = 1 a + 1 b

(a + b)2 = = 1 a2 + 2ab + 1 b2

(a + b)3 = =

(a + b)4 = =

(a + b)5 = =

Leaving out all coefficients in every expansion, we have

Observing the above result, try to expand (a + b)6 without doing the actual multiplication. Can you?

(a + b)6 = ________________________________________________________________________

How about the expansion of (a + b)10? You might wonder whether we should find out the expansion of (a + b)7, (a + b)8, (a + b)9 first. Maybe, it has another way to do that.

Now, evaluate the values of

[pic]

[pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic] [pic]

[pic] [pic] [pic] [pic] [pic]

( Properties of Pascal’s Triangle

1. There are __________ terms in the n th row.

2. Each row begins and ends with a ____.

3. The numbers in the n th row are equal to ______ (r = 0, 1, 2, …, n) respectively.

4. Pascal’s triangle is ______________. (i.e. __________ = _________)

5. The General Term is __________________________

6. Every coefficient of a row can be obtained by adding the two coefficients on its left and right in the row.

(i.e. ______ = ______ + ______)

[[2.1.2 The Binomial Theorem]]

Theorem

After simplifying,

Example Expand

a) (2 + 3x)5

b) ( 1 – x )6

c) (3x – 2y)7

Example Find the 4th term in the expansion of (4x + 5y)10 in descending powers of x.

Example In the expansion of [pic], find

a) the term independent of x,

b) the coefficient of x9.

Example (a) Expand (1 + kx – 2x2)6 in ascending powers of x as far as the term in x2.

(b) If the coefficient of x2 in the above expansion is 123, find the value of k.

Exercise 2A

1. Expand [pic].

2. Expand the following in ascending powers of x as far as the term in x3.

a) (1 + 3x)10

b) ( 1+ x)6( 1- x)4

c) ( 1 + x – 2x2)5

d) ( 1+ 3x + x2)8

e) ( 1 – 4x – 3x2)12

3. Find the term independent of x in the expansion of [pic].

4. Find the constant term in the expansion of [pic].

5. Find the 5th term in the expansion of ( x – 2y)10 in descending powers of x.

6. Find the ratio of the 11 th term and 13 th term in the expansion of ( 3 + x2)15 in ascending powers of x.

7. Simplify [pic].

8. (a) Expand (1 + kx + x2)8 in ascending powers of x as far as the term in x3.

(b) If the coefficient of x2 in the above expansion is 120,

i) find the possible values of k,

ii) find the possible coefficients of x3 in the expansion.

9. Let ( 1+ 5x)2( 1+ x)n = [pic] where n is a positive integer.

a) Express c0, c1, c2 and c3 in terms of n.

b) If c2 = 100, find the value of n.

c) What is the value of the coefficient of x in the expansion of [pic]?

[2.2 The Summation Notation (]

In form 5, we have learnt that [pic]= _________________________________________________.

Example Let x1 = 3, x2 = 5, x3 = -1, x4 = 6. Find the values of the following.

(a) [pic]

(b) [pic]

(c) [pic]

Example Evaluate the following sums.

(a) [pic]

(b) [pic]

(c) [pic]

[[2.2.1 Summation Rules]]

The following properties of the ( notation are useful for later works.

Example Given [pic]= 13, [pic]28 and [pic]= 125, find the values of the following.

(a) [pic]

(b) [pic]

(c) [pic]

• Useful Formulae

[pic]

[pic]

[pic]=

Example Find the sum of [pic].

Exercise 2B

Evaluate the following.

1. [pic]

2. [pic]

3.[pic]

4. [pic]

5. [pic]

6. [pic]

Given [pic], [pic], [pic], [pic] and [pic]= 215. Calculate the following

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. (a) Show that [pic]= [pic].

(b) Prove that (k+1)2 – k2 ( 2k +1.

(c) Using the results in (a) and (b), show that [pic]

d) Hence, or otherwise, find the value of [pic].

[2.3 The Binomial Series]

Expand [pic], we have [pic]= [pic]=

Expand [pic], we have

Expand [pic], we have

For n being a positive integer, we have

(1 + x)n = 1 + [pic]x + [pic]+…. + [pic]+…+xn

= 1 + nx + [pic]x2 + … + xr + … + xn

• The General Binomial Theorem

Indeed, n can be any rational number. But the expansion becomes an infinite series.

You must notify the followings:

When n is an positive integer,

1.[pic]is not limited in any range.

2. The expansion is a finite series.

When n is a rational number,

1. x is limited in (-1,1).

2. The expansion is an infinite series.

Why |x| < 1 such that the expansion is valid?

Let us consider an example to see the reason.

Try to expand (1 + x)-1 by using the general Binomial Theorem.

(1 + x)-1 =

Case I: x = 0.5

L.H.S. = ( 1 + 0.5)-1 = 1.5-1 = [pic]

R.H.S. =

Case II: x = 1

L.H.S. = (1 + 1)-1 = [pic]

R.H.S. =

Case III: x = 1.5

L.H.S. = (1 + 1.5)-1 = [pic]

R.H.S. =

For more details, see or open the excel file.

• Absolute value

|2| may be –2 or 2. Then, |x| =

If a > 0,

|x| < a (

|x| > a (

|x| = a (

Example (a) For |x| < 1,

i) find the coefficient of xr in the expansion of [pic],

ii) write down the expansion of [pic]as far as the term in x4.

(b) Hence, find the value of [pic] correct to 5 significant figures.

[b. 1.0198]

Example Find the expansions of the following in ascending powers of x as far as the term containing x3. In each case, state the range of values of x for which the expansion is valid.

(a) [pic]

(b) [pic]

(c)[pic]

(d) [pic]

Exercise 2C

Use the general binomial theorem, expansion the following expansions up to x3 and state the range of values of x for which the expansion is valid.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. The binomial expansion of (1 + px)-3 in ascending powers of x is 1 + a x + bx2 + [pic]x3 + …, where p , a and b are constants.

a) Find the values of p, a and b.

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

10. Write down the first 3 terms in the expansion of ( 1 – x + 2x2)-3 in ascending powers of x given that the expansion is valid. [Hint: ( 1 – x + 2x2)-3 = [ 1 + (2x2 – x)]-3

11. (a) Expand [pic] in ascending powers of x as far as the term in x3.

(b) State the necessary restriction on the values of x for the expansion in (a).

(c) By taking x = [pic], evaluate [pic] correct to 2 decimal places using the expansion in (a).

12. (a) Find the first 4 terms in the expansion of [pic] in ascending powers of x for –1 < x < 1.

(b) Hence, by taking x = [pic], evaluate [pic] correct to 4 significant figures.

13. (a) If [pic]= [pic] is an identity, find the values of the constants A and B.

(b) Hence, find the expansion of [pic] in ascending powers of x as far as the term in x3.

(c) State for what values of x is the expansion in (b) is valid.

(((((((((((((((((Selected Questions from HKASL MS((((((((((((((((

[1996B-8] There are several bags on a table each containing six cards numbered 0,1,2,3,4 and 5 respectively.

a) (i) Find the coefficient of x5 in the expansion of [pic].

(ii) John takes two bags away from the table and randomly draws a card from each of them. Using (a) (i), or otherwise, find the probability that the sum of the numbers on the two cards drawn is 5.

b) (i) Expand ( 1 – x6)4.

(ii) Find the coefficient of xr, where r is a non-negative integer, in the expansion of ( 1 – x)-4 for |x| < 1.

(iii) Using b(i) and b(ii), or otherwise, find the coefficient of x8 in the expansion of [pic] for |x| < 1.

c) Joan takes four bags away from the table and randomly draws a card from each of them. Using b(iii), or otherwise, find the probability that the sum of the numbers on the four cards drawn is 8.

[1997A-1] Let |ax| < 1.

a) Expand [pic] in ascending powers of x as far as the term in x2.

b) If the coefficient of x2 in the expansion of [pic] is –1, find all possible values of a.

[1998A-2] The binomial expansion of [pic] in ascending powers of x is 1 + bx + cx2 + 160x3 + …, where a, b and c are constants.

a) Find the values of a, b and c.

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

[2000A-2] Let |x| < [pic].

a) Expand [pic] and [pic] respectively in ascending powers of x as far as term in x3.

b) Using (a) and the identity (1 + 2x)(1 – 2x + 4x2) ( 1 + 8x3, or otherwise, expand [pic] in ascending powers of x as far as the term in x3.

[2001A-4] The binomial expansion of [pic] in ascending powers of x is [pic], where a is a constant and n is a positive integer.

a) Find the values of a and n.

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

-----------------------

Distance = 2

0 2

What is |x|?

| | means the absolute value of

e.g. | 2 | = 2

e.g. |-2| = 2

|x| < 1 means the distance between x and 0 is 1

If –1 < x < 1 or |x| < 1 and n is a rational number, then

(1 + x)n = 1 + nx + [pic]+[pic]+…

When n is a +ve integer,

( a + b)n = [pic]

If a and b are constants, then

(a) [pic]

(b) [pic]

When n is a +ve integer,

( a + b)n = [pic]

______________ Triangle

-2

0

0

-1

1

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