Question

what is the 8th term of the binomial expansion ((x + y)^{10})? select the correct answer. (\bigcirc) (120x^{7}y^{3}) (\bigcirc) (720x^{7}y^{3}) (\bigcirc) (120x^{3}y^{7}) (\bigcirc) (720x^{3}y^{7})

Explanation:

Step1: Identify term index formula

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

Step2: Plug in $n=10, k=8$

First calculate $\binom{10}{8-1}=\binom{10}{7}=\frac{10!}{7!(10-7)!}=\frac{10\times9\times8}{3\times2\times1}=120$

Step3: Find exponents of $x$ and $y$

Exponent of $x$: $10-(8-1)=3$, exponent of $y$: $8-1=7$

Step4: Combine components

The term is $120x^3y^7$

Answer:

120$x^3y^7$ (Option C: $120x^3y^7$)