Cambridge Additional Mathematics

Counting and the binomial expansion (Chapter 10) 277

17 The coefficient ofx^2 in the expansion of (1 + 2x)n is 112. Findn.

18 The coefficient ofx^2 in the expansion of

³

1 ¡

x

3

́n

is^53. Findn.

19 The third term of (1 +x)n is 36 x^2. Find the fourth term.

20 Suppose (1 +kx)n=1¡ 12 x+60x^2 ¡::::. Find the values ofkandn.

Review set 10A

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1 Simplify: a

n!

(n¡2)!

b

n!+(n+ 1)!

n!

2 Eight people enter a room and each person shakes hands with every other person. How many hand

shakes are made?

3 The letters P, Q, R, S, and T are to be arranged in a row. How many of the possible arrangements:

a end with T b begin with P and end with T?

4aHow many three digit numbers can be formed using the digits 0 to 9?

b How many of these numbers are divisible by 5?

5 The first two terms in a binomial expansion are: (a+b)^4 =e^4 x¡ 4 e^2 x+::::

a Findaandb. b Copy and complete the expansion.

6 Expand and simplify (

p

3+2)^5 , giving your answer in the form a+b

p

3 , a,b 2 Z.

7 Find the constant term in the expansion of

³

3 x^2 +^1

x

́ 8

.

8 Findcgiven that the expansion (1 +cx)(1+x)^4 includes the term 22 x^3.

9 Steven and nine of his classmates are in a school committee. The committee must select a president,

vice-president, and secretary. In how many ways can this be done if:

a there are no restrictions b Steven must be the president

c Steven cannot hold any of the key positions?

10 Find the coefficient ofx^3 in the expansion of (x+5)^6.

11 A team of five is chosen from six men and four women.

a How many different teams are possible with no restrictions?

b How many different teams contain at least one person of each sex?

12 Find the coefficient ofx¡^6 in the expansion of

³

2 x¡

3

x^2

́ 12

.

13 Find the coefficient ofx^5 in the expansion of (2x+ 3)(x¡2)^6.

14 Find the possible values ofaif the coefficient ofx^3 in

³

2 x+

1

ax^2

́ 9

is 288.

15 In the expansion of (kx¡1)^6 , the coefficient ofx^4 is equal to four times the coefficient ofx^2.

Find the possible values ofk.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_10\277CamAdd_10.cdr Monday, 23 December 2013 4:45:44 PM BRIAN