Question

determine the sixth term of the following expression without expanding the binomial. $(x + y)^9$

Explanation:

Step1: Recall binomial term formula

The $k$-th term of $(a+b)^n$ is $\binom{n}{k-1}a^{n-(k-1)}b^{k-1}$

Step2: Identify values for the formula

Here, $n=9$, $k=6$, $a=x$, $b=y$. Calculate $k-1=5$, $n-(k-1)=9-5=4$

Step3: Compute binomial coefficient

$\binom{9}{5}=\frac{9!}{5!(9-5)!}=\frac{9\times8\times7\times6}{4\times3\times2\times1}=126$

Step4: Assemble the sixth term

Combine coefficient, $x$ term, and $y$ term

Answer:

$126x^4y^5$