Combinatorics & Binomial

4 methods · 11 questions

★ Signature method unlocked: the technique this chapter rewards most often, with 3 worked questions.

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To find the coefficient of a single power, do not expand. Write the general term $(k n) (first)^{n - k} (second)^{k}$ , collect all the powers of the variable into one exponent that is linear in $k$ , set that exponent equal to the target power, and solve the single resulting equation for $k$ . Evaluate that one term only.

Trigger: 'find the coefficient of $x^{r}$ ' (or 'the constant term') in a power of a two-term bracket, even after an outside factor or a divisibility twist.

Instances:
(i) for a bracket with a negative or reciprocal power, the exponent of $x$ is an arithmetic expression in $k$ (such as $72 - 8 k$ or $6 - 2 k$ ); set it to the target and read off $k$ ;
(ii) if the bracket is multiplied by a power of $x$ first, subtract that shift from the target power before solving for $k$ ;
(iii) for a divisibility or sign question, keep the term in the symbolic form $(k n) a^{n - k} b^{k}$ and reason on its factors rather than its value.

Linked questions (3)

TMUA 2018 Paper 1 · Q7

The non-zero constant

k

is chosen so that the coefficients of

x^{6}

in the expansions of

(1 + k x^{2})^{7}

and

(k + x)^{10}

are equal. What is the value of

k

A $\frac{1}{6}$
B $6$
C $\frac{6}{6}$
D $6$
E $\frac{30}{30}$
F $30$

TMUA 2018 Paper 2 · Q2

Find the value of the constant term in the expansion of

(x^{6} - \frac{1}{x ^{2}})^{12}

A $- 495$
B $- 220$
C $- 66$
D $66$
E $220$
F $495$

TMUA 2019 Paper 2 · Q1

Find the coefficient of the

x^{4}

term in the expansion of

x^{2} (2 x + \frac{1}{x})^{6}

A15
B30
C60
D120
E240