0604 Binomial Coefficients

p436

THE BINOMIAL THEOREM Let $x$ and $y$ be variables, and let n be a nonnegative integer. Then

$\begin{aligned} (x+y)^n &= \binom{n}{0}x^ny^0 + \binom{n}{1}x^{n-1}y^1 + \cdots + \binom{n}{n}x^0y^n\\ &=\sum^n_{k=0}x^{n-k}y^k \end{aligned}$

1-4

p443: 4, 7, 8, 9

4. Find the coefficient of $x^5y^8$ in $(x + y)^{13}$ .

Solution

$(x+y)^{13} = \sum^{13}_{k=0}\binom{13}{k}x^{13-k}y^k$

The coefficient of $x^5y^8$ is $\tbinom{13}{8} = \frac{13!}{5!8!}$ .

7. What is the coefficient of $x^9$ in $(2-x)^{19}$ ?

Solution

$(2-x)^{19} = \sum^{19}_{k=0}\binom{19}{k}2^{19-k}(-x)^k$

The coefficient of $x^9$ is $\tbinom{19}{9} \cdot 2^{10} \cdot (-1)^9= -\frac{2^9\cdot 19!}{10!9!}$ .

8. What is the coefficient of $x^8y^9$ in the expansion of $(3x + 2y)^{17}$ ?

Solution

$(3x+2y)^{17} = \sum^{17}_{k=0}\binom{17}{k}(3x)^{17-k}(2y)^k$

The coefficient of $x^8y^9$ is $\tbinom{17}{9} \cdot 3^8 \cdot 2^9= \frac{3^8 \cdot 2^9 \cdot 17!}{8!9!}$ .

9. What is the coefficient of $x^{101}y^{99}$ in the expansion of $(2x-3y)^{200}$ ?

Solution

$(2x-3y)^{200} = \sum^{200}_{k=0}\binom{200}{k}(2x)^{200-k}(-3y)^k$

The coefficient of $x^{101}y^{99}$ is $\tbinom{200}{99} \cdot 2^{101} \cdot (-3)^{99}= -\frac{2^{101} \cdot 3^{99} \cdot 200!}{101!99!}$ .