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