Question

1. which is the value of the 3rd term in the expansion of ((x + 6)^6)?

(\boldsymbol{dbinom{6}{2} 36x^4})
(\boldsymbol{dbinom{6}{4} 36x^2})
(\boldsymbol{dbinom{6}{4} 36x^4})
(\boldsymbol{dbinom{6}{2} 36x^2})

Explanation:

Step1: Recall binomial term formula

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

Step2: Identify values for the 3rd term

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

Step3: Calculate the term components

$\binom{6}{2}x^{6-2}6^{2} = \binom{6}{2}x^4 \cdot 36$. Note $\binom{6}{2}=\binom{6}{4}$, so this can also be written as $\binom{6}{4}36x^4$.

Answer:

$\boldsymbol{\dbinom{6}{2} 36x^4}$ (equivalent to $\boldsymbol{\dbinom{6}{4} 36x^4}$)