Question

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

Step1: Recall binomial term formula

For $(a+b)^n$, the $k$-th term is $\binom{n}{k-1}a^{n-(k-1)}b^{k-1}$

Step2: Identify values for the problem

Here, $a=2x$, $b=3y$, $n=14$, $k=7$. So $k-1=6$

Step3: Calculate binomial coefficient

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

Step4: Compute $a$ component

$a^{14-6}=(2x)^8=2^8x^8=256x^8$

Step5: Compute $b$ component

$b^{6}=(3y)^6=3^6y^6=729y^6$

Step6: Multiply all components

$3003\times256\times729x^8y^6$
First calculate $3003\times256=768768$, then $768768\times729=550431872$

Answer:

$550431872x^8y^6$