# What is the coefficient of x^101 y^99 in the expansion of (2x - 3y)^200? A. C(200, 99) 2^101 (3)^99 B. C(200, 99) 2^101(-3)^99 C. P(200, 99) 2^101(3)^99 D. P(200, 99) 2^101(-3)^99 E. C(200, 2) 2^101(-3)^99

What is the coefficient of ${x}^{101}{y}^{99}$ in the expansion of $\left(2x-3y{\right)}^{200}$?
A. $C\left(200,99\right){2}^{101}\left(3{\right)}^{99}$
B. $C\left(200,99\right){2}^{101}\left(-3{\right)}^{99}$
C. $P\left(200,99\right){2}^{101}\left(3{\right)}^{99}$
D. $P\left(200,99\right){2}^{101}\left(-3{\right)}^{99}$
E. $C\left(200,2\right){2}^{101}\left(-3{\right)}^{99}$
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Solution:
Given expansion $\left(2x-9y{\right)}^{200}$ the coefficient of ${x}^{101}{y}^{99}$
$\left(x+a{\right)}^{n}$ Generate term ${T}_{\gamma +1}={n}_{{r}_{\gamma }}{x}^{n-\gamma }{a}^{\gamma }$
$={200}_{{c}_{\gamma }}{2}^{200-\gamma }{x}^{200-\gamma }\left(-3{\right)}^{\gamma }{y}^{\gamma }$
here r=99, 200-99=101
$={200}_{{c}_{9}9}{2}^{200-99}{x}^{200-99}\left(-3{\right)}^{99}{y}^{99}$
$={200}_{{c}_{9}9}{2}^{101}{x}^{101}\left(-3{\right)}^{99}{y}^{99}$
The coefficient of ${x}^{101}{y}^{99}$ is
${200}_{{c}_{9}9}{2}^{101}\left(-3{\right)}^{99}$
option B