Answer:
The correct option is 1. The given statement is true.
Step-by-step explanation:
The binomial expansion is defined as
[tex](x+y)^n=^nC_0x^{n-0}y^{1}+^nC_{1}x^{n-1}y^{2}+....+^nC_{n-1}x^{1}y^{n-1}+^nC_nx^{0}y^{n}[/tex]
The rth term in a binomial expansion is defined as
[tex]\text{rth term}=^nC_rx^{n-r}y^{r}[/tex]
Let the coefficient of [tex]x^ky^{n-k}[/tex] be A. The power of x is k and the power of y is n-k. It means
[tex]k=n-r[/tex]
[tex]n-k=r[/tex]
The coefficient of [tex]x^ky^{n-k}[/tex] is
[tex]^nC_{n-k}=\binom{n}{n-k}[/tex]
Using the property of combination,
[tex]^nC_{n-r}=^nC_r[/tex]
[tex]^nC_{n-k}=^nC_{k}=\binom{n}{k}[/tex]
The coefficient of [tex]x^ky^{n-k}[/tex] is [tex]\binom{n}{k}[/tex]. Therefore the given statement is true.
