Question

determine the seventh term of the following expression without expanding the binomial. $(2x + 3y)^{14}$

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, $n=14$, $k=7$, $a=2x$, $b=3y$. Calculate $k-1=6$, $n-(k-1)=14-6=8$.

Step3: Compute binomial coefficient

$\binom{14}{6}=\frac{14!}{6!(14-6)!}=3003$

Step4: Calculate $a$ and $b$ components

$(2x)^8=2^8x^8=256x^8$, $(3y)^6=3^6y^6=729y^6$

Step5: Multiply all components

$3003 \times 256 \times 729 \times x^8y^6$
First compute $3003 \times 256=768768$, then $768768 \times 729=550431872$

Answer:

$550431872x^8y^6$