Chapter 1 Permutations and Combinations

Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note 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

______Triangle

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

1 1 C0 C1

2 2 2 C0 C1 C2

Due Date: ______Received Date: ______Ch2-1 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

3 3 3 3 C0 C1 C2 C3

4 4 4 4 4 C0 C1 C2 C3 C4

 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 When n is a +ve integer, n n n0 0 n n1 1 n n2 2 n nr r n nn n ( a + b) = C0 a b  C1 a b  C2 a b  ...  Cr a b  ...  Cn a b

After simplifying,

When n is a +ve integer, n n n n1 1 n n2 2 n nr r n ( a + b) = a  C1 a b  C2 a b  ...  Cr a b  ...  b

Example 1 Expand (a) (2 + 3x)5 (b) ( 1 – x )6 (c) (3x – 2y)7

Example 2 Find the 4th term in the expansion of (4x + 5y)10 in descending powers of x.

12  2  Example 3 In the expansion of  x 2   , find  x 

(a) the term independent of x, (b) the coefficient of x9.

Example 4 (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 (1 2x 2 )7 .

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

9  3  3. Find the term independent of x in the expansion of 2x   .  x 2 

8  4  4. Find the constant term in the expansion of  x   .  x 3 

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 (x  2)5  (x  2)5 .

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. 2 n 2 3 Let ( 1+ 5x) ( 1+ x) = c0  c1 x  c2 x  c3 x  .... 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 (1 5x) 2 (1 x) n1 ?

[2.2 The Summation Notation ] n In form 5, we have learnt that  xi = ______. i1

Example 5 Let x1 = 3, x2 = 5, x3 = -1, x4 = 6. Find the values of the following. 4 (a)  xi i1 4 (b) (7xi  2) i1 3 2 (c)  xi i1

Example 6 Evaluate the following sums. 100 (a)  k k 1 7 (b) (2n  6) n4 4 t (c)  t2 t 1

[[2.2.1 Summation Rules]] The following properties of the  notation are useful for later works.

If a and b are constants, then n (a) (axi  byi )  i1 n (b) (axi  b)  i1

9 9 9 2 Example 7 Given  xi = 13,  yi  28 and  xi = 125, find the values of the following. i1 i1 i1 9 (a) (2xi  5yi ) i1 9 (b) (4yi  6) i1 9 2 (c) (xi  3) i1

 Useful Formulae n  r  r1 n  r 2  r1 n  r 3 = r1

50 Example 8 Find the sum of  r(r 1)(r  2) . r1

Exercise 2B Evaluate the following. 10 1. (2t 1) t0 30 2.  n3 n4 Due Date: ______Received Date: ______Ch2-8 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

5 j 3.  2 j1 6 6 4. Cr r0 7 7 5.  Pk k3 3 1 6.  m1 m(m 1) 5 5 5 5 5 2 2 Given  xi  31 ,  yi  28 ,  xi  255 ,  yi  186 and  xi yi = 215. Calculate the following i1 i1 i1 i1 i1 5 7. (2xi  7) i1 5 8. (3xi  4yi ) i1 5 2 9. (xi  4) i1 5 2 10. (8  3yi ) i1 5 2 11. (xi  yi ) i1 n 12. (a) Show that (xk 1  xk ) = xn1  x1 . k 1 (b) Prove that (k+1)2 – k2  2k +1. n 1 (c) Using the results in (a) and (b), show that  k  n(n 1) k 1 2 50 (d) Hence, or otherwise, find the value of (3k  5) . k17

[2.3 The Binomial Series] n! Expand C n , we have C n = = 1 1 1!(n 1)! n Expand C2 , we have

n Expand Cr , we have

For n being a positive integer, we have

n n n 2 n r n (1 + x) = 1 + C1 x + C2 x +…. + Cr x +…+x n(n 1) = 1 + nx + x2 + … + xr + … + xn 2

 The General Binomial Theorem Due Date: ______Received Date: ______Ch2-10 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

Indeed, n can be any rational number. But the expansion becomes an infinite series. If –1 < x < 1 or |x| < 1 and n is a rational number, then n(n 1) n(n 1)(n  2) n(n 1)    (n  r 1) (1 + x)n = 1 + nx + x 2  x3  ... + x r +… 2! 3! r!

You must notify the followings: What is |x|? When n is an positive integer, | | means the absolute value of e.g. | 2 | = 2 1. x is not limited in any range. 2. The expansion is a finite series. When n is a rational number, e.g. |-2| = 2 1. x is limited in (-1,1). 0 2 2. The expansion is an infinite series. Distance = 2 |x| < 1 means the distance between x and 0 is 1 Why |x| < 1 such that the expansion is valid? Let us consider an example to see the reason. -2 0 Try to expand (1 + x)-1 by using the general Binomial Theorem.

-1 (1 + x) = -1 0 1

Case I: x = 0.5 2 L.H.S. = ( 1 + 0.5)-1 = 1.5-1 = 3 Due Date: ______Received Date: ______Ch2-11 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

R.H.S. =

Case II: x = 1 1 L.H.S. = (1 + 1)-1 = 2 R.H.S. =

Case III: x = 1.5 2 L.H.S. = (1 + 1.5)-1 = 5 R.H.S. =

For more details, see http://www.mrsiu.net/ 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 9 (a) For |x| < 1,

1 r (i) find the coefficient of x in the expansion of (1 x) 2 ,

1 4 (ii) write down the expansion of (1 x) 2 as far as the term in x . (b) Hence, find the value of 1.04 correct to 5 significant figures.

[b. 1.0198]

Example 10 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) (1 x) 2

1 (b) (1 2x) 3

1 3 (c) (1 2x) (1 x)2

(d) (2  x) 1

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. Due Date: ______Received Date: ______Ch2-13 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

1. (1 x) 3 1 2. (1 x) 2

1 3. (1 x) 2

1  4. (1 x) 3

5. (1 x) 4

1 6. (1 x) 5

1 7. (1 3x) 2

8. (x  2)1

16 9. The binomial expansion of (1 + px)-3 in ascending powers of x is 1 + a x + bx2 + x3 + …, where p , a and b are 25 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

1 11. 3 (a) Expand (1 2x) 3 in ascending powers of x as far as the term in x .

(b) State the necessary restriction on the values of x for the expansion in (a). 1 (c) By taking x = , evaluate 3 6 correct to 2 decimal places using the expansion in (a). 8

1 x 12. (a) Find the first 4 terms in the expansion of in ascending powers of x for –1 < x < 1. 1 x 1 (b) Hence, by taking x = , evaluate 5 correct to 4 significant figures. 9

7  2x A B 13. (a) If =  is an identity, find the values of the constants A and B. (1 x)(2  x) 1 x 2  x 7  2x (b) Hence, find the expansion of in ascending powers of x as far as the term in x3. (1 x)(2  x) (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 (1 x  x 2  x 3  x 4  x 5 ) 2 . (ii) John takes two bags away from the table and randomly draws a card from each of them. Using (a) (i), or Due Date: ______Received Date: ______Ch2-16 Name: ______Class: ______Class No.: ______Marks: ______S6 MS Chapter 2 Note

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.

6 4 8 1 x  (iii) Using b(i) and b(ii), or otherwise, find the coefficient of x in the expansion of   for |x| < 1.  1 x 

(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.

1 2 (a) Expand (1 ax) 3 in ascending powers of x as far as the term in x .

1 2 (b) If the coefficient of x in the expansion of (1 ax) 3 is –1, find all possible values of a.

[1998A-2] The binomial expansion of (1 ax) 4 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.

1 [2000A-2] Let |x| < . 2

1 1  3 (a) Expand (1 2x) 2 and (1 8x 3 ) 2 respectively in ascending powers of x as far as term in x .

1 2 3  (b) Using (a) and the identity (1 + 2x)(1 – 2x + 4x )  1 + 8x , or otherwise, expand (1 2x  4x 2 ) 2

in ascending powers of x as far as the term in x3.

1  4 32 2 [2001A-4] The binomial expansion of n in ascending powers of x is 1 x  x  ..., where a is a (1 ax) 3 9 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.

Due Date: ______Received Date: ______Ch2-18