First slide

Multinomial expansion

Question

$The coefficient of x^{2} y^{3} z^{4} in (a x - b y + c z)^{9} is$

Moderate

A

$1260 a^{2} b^{3} c^{4}$
B

$- 1260 a^{2} b^{3} c^{4}$
C

$1220 a^{2} b^{3} c^{4}$
D

$- 1220 a^{2} b^{3} c^{4}$

Solution

$In the expansion of (a x - b y + c z)^{9}, the term containing z^{4} {is}^{9} C_{4} {(ax-by)}^{5} {(cz)}^{4}$
$\begin{array}{l} In the expansion of (a x - b y)^{5}, the term containing y^{3} {is}^{5} C_{3} (a x)^{2} (- b y)^{3} \\ ∴ The coefficient of x^{2} y^{3} z^{4} {is}^{9} C_{4} [^{5} C_{3} a^{2} (- b)^{3}] (c)^{4} = \\ ^{9} C_{4} [^{5} C_{3} a^{2} (- b)^{3}] (c)^{4} \frac{9 \times 8 \times 7 \times 6}{24} \times 10 \times a^{2} b^{3} c^{4} = - 1260 a^{2} b^{3} c^{4} \end{array}$

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