Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

De term in x⁹ van $https://latex.codecogs.com/png.image?\dpi{120}\left ( x^3-\frac{1}{2\, \sqrt{x}} \right )^{10}$ heeft een coëfficiënt die gelijk is aan	A. \(\boldsymbol{\frac {105} {32} }\)
B. \(\boldsymbol{\frac {15} {2} }\)
C. \(\boldsymbol{-\,\frac {105} {32} }\)
D. \(\boldsymbol{\frac {105} {4} }\)
E. \(\boldsymbol{{105} }\)
F. \(\boldsymbol{\frac {105} {8} }\)

IN CONSTRUCTION	A.
B.
C.
D.
E.

Wegens de formule $(a+b)^n=\sum_{k=0}^{n}C_n^k.\,a^{n-k}.\,b^k$ is $\\(x^3-\frac{1}{2\sqrt x})^{20}=\sum_{k=0}^{10}C_{10}^k.\,(x^3)^{10-k}.\,\left(-\frac{1}{2\sqrt x}\right)^k=\sum_{k=0}^{10}C_{10}^k\left(-\frac12\right)^k(x^3)^{10-k}\left(\frac{1}{\sqrt x}\right)^k\\=\sum_{k=0}^{10}C_{10}^k\left(-\frac12\right)^kx^{30-3k}x^{-\frac k2}=\sum_{k=0}^{10}C_{10}^k(-1)^k2^{-k}x^{30-3k-\frac k2}$
De eis die gesteld wordt is : $30-3k-\frac k2=9\;\Leftrightarrow\;21=3.5k\;\Leftrightarrow\;42=7k\;\Leftrightarrow\;k=6$
k ∈ { , 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } zodat de bewuste term kan berekend worden :
$\\C_{10}^6\,(-1)^6(-2)^{-6}x^9=C_{10}^4\:2^{-6}x^9=\frac{10.9.8.7}{2.3.4.2^6}x^9=\frac{10.3.7}{2^6}x^9=\frac{210}{64}x^9=\frac{105}{32}x^9$