Solution: 2022 Winter Final - 6

Author: Michiel Smid

Question

What is the coefficient of $x^{10}y^{20}$ in the expansion of $(2x - 3y)^{30}$?

(a)

$\binom{30}{20}\cdot 2^{10}\cdot 3^{20}$

(b)

$\binom{30}{20}\cdot 2^{20}\cdot 3^{10}$

(c)

$\binom{30}{20}\cdot 2^{30}\cdot 3^{10}$

(d)

$-\binom{30}{10}\cdot 2^{20}\cdot 3^{10}$

(e)

$\binom{30}{10}\cdot 2^{20}\cdot 3^{10}$

$ (2x - 3y)^{30} $

$ = \sum_{k=0}^{30} \binom{30}{k} \cdot (2x)^{n-k} \cdot (-3y)^{k} $

We only consider $k=20$, as it results in $y^{20}$.

$ = \binom{30}{20} \cdot (2x)^{30-20} \cdot (-3y)^{20} $

$ = \binom{30}{20} \cdot (2x)^{10} \cdot (-3y)^{20} $

$ = \binom{30}{20} \cdot 2^{10} \cdot (-3)^{20} \cdot x^{10} \cdot y^{20} $

$ = \binom{30}{20} \cdot 2^{10} \cdot (3)^{20} \cdot x^{10} \cdot y^{20} $

From this equation, we can see that the coefficient (aka the real numbers) are: $\binom{30}{20} \cdot 2^{10} \cdot 3^{20}$