what is the coefficient of the $x^{4}$-term in the binomial expansion of $(x + 3)^{12}$?\n$_{12}c_{7}(3)^{7}$…

what is the coefficient of the $x^{4}$-term in the binomial expansion of $(x + 3)^{12}$?\n$_{12}c_{7}(3)^{7}$\n$_{12}c_{8}$\n$_{12}c_{4}(3)^{4}$\n$_{12}c_{8}(3)^{8}$

what is the coefficient of the $x^{4}$-term in the binomial expansion of $(x + 3)^{12}$?\n$_{12}c_{7}(3)^{7}$\n$_{12}c_{8}$\n$_{12}c_{4}(3)^{4}$\n$_{12}c_{8}(3)^{8}$

Answer

Answer:

D. ${12}C{8}(3)^{8}$

Explanation:

Step1: Recall binomial - theorem formula

The binomial theorem states that $(a + b)^n=\sum_{k = 0}^{n}{{n}C{k}a^{n - k}b^{k}}$. In the expansion of $(x + 3)^{12}$, we have $a=x$, $b = 3$, and $n = 12$. So, the general term is $T_{k+1}={{12}C{k}x^{12 - k}(3)^{k}}$.

Step2: Find the value of k for the $x^{4}$ - term

We want to find the coefficient of the $x^{4}$ - term. Set $12−k = 4$. Solving for $k$, we get $k=12 - 4=8$.

Step3: Substitute k into the general - term formula

When $k = 8$, the term is $T_{9}={{12}C{8}x^{12 - 8}(3)^{8}}={{12}C{8}x^{4}(3)^{8}}$. So the coefficient of the $x^{4}$ - term is ${12}C{8}(3)^{8}$.