Binomial Theorem practice questions

Secondary 3 topics from the Additional Mathematics 4049 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise Binomial Theorem free

What this topic covers

Applications of the Binomial Theorem
Binomial Coefficients and Factorial Notation
Finding Specific Terms
The Binomial Theorem

Worked examples

Example 1

Find the value of

n

given that

\binom{n}{2} = 66

Show the worked solution

$\binom{n}{2} = \frac{n(n-1)}{2}$
$\frac{n(n-1)}{2} = 66$
$n(n-1) = 132$
$n^2 - n - 132 = 0$
$(n - 12)(n + 11) = 0$
$n = 12 \quad \scriptsize\textit{since }n > 0$

Answer: $12$

Example 2

Find the coefficient of

x^{3}

in the expansion of

(2 - 3x)^{6}

Show the worked solution

$T_{4} = \binom{6}{3} \cdot 2^{3} \cdot (-3x)^{3} \quad \scriptsize\textit{general term with } r = 3$
$= 20 \times 8 \times (-27)x^{3}$
$= -4320x^{3}$
Coefficient of $x^{3} = -4320$

Answer: $-4320$

Example 3

Find the coefficient of

x^{6}

in the expansion of

\left(3x + \frac{3}{x}\right)^{8}

Show the worked solution

General term: $T_{r+1} = \binom{8}{r}(3x)^{8-r}\left(\frac{3}{x}\right)^{r}$
Power of $x = (8 - r) - r$
$= 8 - 2r$
For the coefficient of $x^{6}$ : $8 - 2r = 6$
$2r = 2$
$r = 1$
Coefficient $= \binom{8}{1} \times 3^{7} \times 3^{1}$
$= 8 \times 2187 \times 3$

Answer: $52488$

Questions parents ask

Is it really free?

The free plan is permanent and does not ask for a card. It has daily limits on the AI feedback, and the paid plans lift them.

Which syllabus does this follow?

Additional Mathematics (4049), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

What this topic covers

Worked examples

Example 1

Example 2

Example 3

Questions parents ask

Is it really free?

Which syllabus does this follow?

Will it just show my child the answer?

Other Secondary 3 Additional Math topics