Find the 4th term in the binomial expansion. $$( z + y ) ^ { n }$$
A) $\frac { ( n - 3 ) ! } { ( n - 3 ) ! 3 ! } z ^ { n - 3 } y ^ { 3 }$
B) $\frac { n ! } { ( n - 4 ) ! } z ^ { n - 4 } y ^ { 4 }$
C) $\quad \frac { n ! } { ( n - 3 ) ! 3 ! } z ^ { n - 3 } y ^ { 3 }$
D) $\frac { n ! } { ( n - 4 ) ! 4 ! } z ^ { n - 4 } y ^ { 4 }$
E) $\frac { n ! } { ( n - 3 ) ! 3 ! } z ^ { n - 3 } y ^ { n }$

Question

Quizplus · Accepted Answer

The general formula for the term $k$ in the binomial expansion of $(z+y)^n$ is $\binom{n}{k}z^{n-k}y^k$. In this case, we want the 4th term, which means $k=3$. Plugging in $k=3$ gives us $\binom{n}{3}z^{n-3}y^3$. Simplifying the binomial coefficient using $\binom{n}{k}=\frac{n!}{k!(n-k)!}$, we get $\frac{n!}{3!(n-3)!}z^{n-3}y^3$, which matches choice C.

Find the 4th Term in the Binomial Expansion (z+y)n( z + y ) ^ { n }(z+y)n

Find the 4th Term in the Binomial Expansion $( z + y ) ^ { n }$