what is the fifth term in the binomial expansion of (x + 5)^8?\n175,000x^3\n43,750x^4\n3,125x^5\n7,000x^5

Explanation:

Step1: Recall binomial - theorem formula

The general term in the binomial expansion of ((a + b)^n) is given by (T_{r + 1}=C(n,r)a^{n - r}b^{r}), where (C(n,r)=\frac{n!}{r!(n - r)!}), (n) is the power of the binomial, (r) is the term - number minus 1. Here (n = 8), (a=x), (b = 5), and we want the fifth term, so (r+1 = 5), then (r = 4).

Step2: Calculate the binomial coefficient (C(8,4))

[ \begin{align*} C(8,4)&=\frac{8!}{4!(8 - 4)!}\ &=\frac{8!}{4!4!}\ &=\frac{8\times7\times6\times5}{4\times3\times2\times1}\ &=70 \end{align*} ]

Step3: Find the fifth term

[ \begin{align*} T_{5}&=C(8,4)x^{8 - 4}\times5^{4}\ &=70\times x^{4}\times625\ &=43750x^{4} \end{align*} ]

Answer:

43,750(x^{4})