StudySmarter AI is coming soon!

- :00Days
- :00Hours
- :00Mins
- 00Seconds

A new era for learning is coming soonSign up for free

Suggested languages for you:

Americas

Europe

Q9E

Expert-verifiedFound in: Page 421

Book edition
7th

Author(s)
Kenneth H. Rosen

Pages
808 pages

ISBN
9780073383095

**${{\mathit{x}}}^{{\mathbf{101}}}{{\mathit{y}}}^{{\mathbf{99}}}$What is the coefficient of **** in the expansion of ${\mathbf{(}}{\mathbf{2}}{\mathit{x}}{\mathbf{-}}{\mathbf{3}}{\mathit{y}}{\mathbf{)}}^{\mathbf{200}}{\mathbf{?}}$**

The coefficient ${x}^{101}{y}^{99}\text{is then}-\frac{200!}{99!(200-99)!}{2}^{101}{3}^{99}\approx -39\times {10}^{135}$

**Binomial theorem: binomial theorem, statement that for any positive integer n ****, the nth power of the sum of two numbers x ****and y**** may be expressed as the sum of **** terms of the form.**

${\mathbf{(}}{\mathit{x}}{\mathbf{+}}{\mathit{y}}{\mathbf{)}}^{\mathbf{n}}{\mathbf{=}}\mathbf{\sum}_{\mathbf{j}\mathbf{=}\mathbf{0}}^{\mathbf{n}}{\mathbf{\u200a}}{\left(\begin{array}{l}n\\ j\end{array}\right)}{{\mathit{x}}}^{\mathbf{n}\mathbf{-}\mathbf{j}}{{\mathit{y}}}^{{\mathbf{j}}}$

The term is ${x}^{101}{y}^{99}\text{in}(2x-3y{)}^{200}=(2x+(-3y){)}^{200}$

$\begin{array}{r}n=200\\ j=99\end{array}$

$\begin{array}{r}\left(\begin{array}{c}n\\ j\end{array}\right)\left(2x{)}^{n-j}\right(-3y{)}^{j}=\left(\begin{array}{c}200\\ 99\end{array}\right)\left(2x{)}^{200-99}\right(-3y{)}^{99}\\ =-\frac{200!}{99!(200-99)!}\left(2x{)}^{101}\right(3y{)}^{99}\\ =-\frac{200!}{99!(200-99)!}{2}^{101}{x}^{101}{3}^{99}{y}^{99}\\ =-\frac{200!}{99!(200-99)!}{2}^{101}{3}^{99}{x}^{101}{y}^{99}\\ \left(\begin{array}{l}n\\ j\end{array}\right)\left(2x{)}^{n-j}\right(-3y{)}^{j}\approx -39\times {10}^{135}{x}^{101}{y}^{99}\end{array}$

Thus, the coefficient ${x}^{101}{y}^{99}\text{is then}-\frac{200!}{99!(200-99)!}{2}^{101}{3}^{99}\approx -39\times {10}^{135}$

94% of StudySmarter users get better grades.

Sign up for free