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5.09 Pascal's triangle and binomial expansion

Worksheet
Pascal's triangle
1

The 4th row of Pascal’s triangle consists of the numbers 1, 4, 6, 4, 1.

a

Write down the numbers in the 5th row of Pascal’s triangle.

b

Find the sum of the entries in the 5th row of Pascal’s triangle.

c

Rewrite each element of the 5th row of Pascal’s triangle as an {}^{n}C_{r}coefficient.

2

Complete the following entries in a particular row of Pascal’s triangle: 1,8,⬚,56,70,⬚, 28,⬚,1

3

Which row of Pascal’s triangle would give us the coefficients of terms in the expansion of \left(x + y\right)^{7}?

4

State the number of terms in the following expansions:

a

\left( 4 x + y\right)^{6}

b

\left(m + y\right)^{8}

5

Consider the expansion of \left(y + k\right)^{4}.

a

How many terms will there be in the expansion?

b

Write the coefficient of the 2nd term.

6

Using the relevant row of Pascal’s triangle, determine the coefficient of each term in the following expansion:

\left(x + m\right)^{4} = ⬚ x^{4} m^{0} + ⬚ x^{3} m^{1} + ⬚ x^{2} m^{2} + ⬚ x^{1} m^{3} + ⬚ x^{0} m^{4}

Binomial theorem
7

Evaluate the following:

a

\binom{6}{5}

b

\binom{14}{1}

c

\binom{10}{9}

d

\binom{8}{0}

8

Find the specified term for the following expansions:

a

Fifth term of \left(b + 2\right)^{7}.

b

Fifth term of \left(c + d\right)^{9}.

c

Fifth term of (3u-v)^{6}.

d

Tenth term of \left(u + \dfrac{1}{2}\right)^{12}.

e

Seventh term of \left(\dfrac{3 y}{2} - \dfrac{2}{3 y}\right)^{10}.

f

Eleventh term of \left( 2 x + y^{2}\right)^{13}.

g

Sixteenth term of \left(x - y^{5}\right)^{19}.

h

Middle term of \left( 4 x^{2} + 3 y^{3}\right)^{8}.

i

Middle term of \left( - 3 x^{ - 3 } + 2 y^{ - 2 }\right)^{4}.

j

Constant term of \left( 2 y - \dfrac{1}{y^{3}}\right)^{8}.

9

The first three terms of \left(x + y\right)^{10} are x^{10}, 10 x^{9} y and 45 x^{8} y^{2}. Using the symmetry of the coefficients, write the last three terms of the expansion.

10

In the expansion of \left(m + k\right)^{n}, the coefficient of the 9th term is 45.

a

Find the value of n.

b

Find the 9th term.

11

A particular term in the expansion of \left( 3 a^{2} + \dfrac{p}{b}\right)^{4} is \dfrac{96 a^{2}}{b^{3}}, for some constant p. Find the value of p.

12

Consider the expansion of \left(a - 4\right)^{5}. Will the 2nd term be positive or negative?

13

By considering \left(x + y\right)^{4} = \left(x + y\right) \left(x + y\right) \left(x + y\right) \left(x + y\right), in how many ways can three factors of x and one factor of y be chosen from the expansion to form the term which contains x^{3} y^{1}? Write your answer in the form {}^{n}C_{r}.

14

Complete the following expansion:

\left( 4 p + 3 q\right)^{3} = {}^{3}C_{0} \left( 4 p\right)^{⬚} \left( 3 q\right)^{0} + {}^{3}C_{1} \left( 4 p\right)^{⬚} \left( 3 q\right)^{1} + {}^{3}C_{2} \left( 4 p\right)^{⬚} \left( 3 q\right)^{2} + {}^{3}C_{3} \left( 4 p\right)^{⬚} \left( 3 q\right)^{3}

15

Consider the binomial \left( 4 x + 3 y\right)^{4}.

a

State the first term in the expansion.

b

State the last term in the expansion.

16

Consider the expansion of \left(a + b\right)^{5}.

a

State the coefficient of the 3rd term in the form {}^{n}C_{r}.

b

Evaluate the coefficient of the 3rd term.

c

By considering the symmetry of {}^{n}C_{r}, which other term has the same coefficient as the 3rd term?

17

Which two terms in the expansion of \left(u + v\right)^{11} have a coefficient of {}^{11}C_{9}?

18

Find the coefficient of x^{17} in the expansion of \left( 3 x^{2} + 2 x\right)^{11}.

19

Use the binomial theorem to expand the following expressions:

a

\left(1 + x\right)^{4}

b

\left(2 + b\right)^{5}

c

\left(p + q\right)^{5}

d

\left(c - d\right)^{8}

e

\left(u + 4 v\right)^{6}

f

\left(y + 3\right)^{4}

g

( k a - b)^{6}

h

\left(y - 5\right)^{3}

i

\left(y - \dfrac{1}{3}\right)^{3}

j

\left(2 + \dfrac{y}{2}\right)^{3}

k

\left( 3 y + 2 r\right)^{3}

l

\left(a - \dfrac{1}{a}\right)^{3}

m

\left(u^{2} + 3 v^{2}\right)^{3}

n

\left(1 + \dfrac{4}{y}\right)^{3}

20

Find:

a

The term in \left(3 + \sqrt{x}\right)^{10} that contains x^{4}.

b

The term in \left(c + d\right)^{10} that contains c^{3}.

c

The term in \left(u + v\right)^{11} that contains v^{8}.

d

The term in \left(u + v\right)^{15} that contains u^{5}.

21

If the eighth and tenth terms in the expansion of \left(x + y\right)^{n} have the same coefficients, find the value of n.

22

Find the coefficient of a^{3} b^{6} in the expansion of \left( 2 a - \dfrac{b}{2}\right)^{9}.

23

Consider the expansion of \left(1 + x\right)^{9}. State the coefficient of the 4th term in the form {}^{n}C_{r}.

24

Find the value of n such that the expansion of \left(a + b\right)^{n} contains the term 84 a^{6} b^{3}.

25

Complete the following expansion by determining the missing power and binomial coefficient of the \left(k + 1\right)th term:

\left(3 - x\right)^{8} = {}^{8}C_{0} \times 3^{8} \left( - x \right)^{0} + {}^{8}C_{1} \times 3^{7} \left( - x \right)^{1}+\ldots+⬚ \times 3^{8 - k} \left( - x \right)^{⬚}+\ldots+{}^{8}C_{8} \times 3^{0} \left( - x \right)^{8}

26

Consider the binomial series:

\left(1 + x\right)^{n} = 1 + n x + \dfrac{n \left(n - 1\right)}{2!} x^{2} + \dfrac{n \left(n - 1\right) \left(n - 2\right)}{3!} x^{3} + \ldots

a

Use the first four terms to approximate \left(1.03\right)^{ - 2 } to the nearest thousandth.

b

Use the first four terms to approximate \left(1.05\right)^{\frac{3}{4}} to the nearest thousandth.

c

Use the first five terms to approximate \dfrac{1}{\left(1.06\right)^{2}} to the nearest thousandth.

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Outcomes

1.3.4.2

recognise and use the link between Pascal’s triangle and the notation (n r)

1.3.4.3

expand (𝑥+𝑦)^𝑛 for small positive integers n

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