Question

If $n$ and $r$ are two positive integers such that $n \geq r$, then ${ }^{n} C_{r-1}+{ }^{n} C_{r}$ is equal to:

• A. ${ }^{n} C_{r-1}$
• B. ${ }^{n} C_{r}$
• C. ${ }^{n-1} C_{r}$
• D. ${ }^{n+1} C_{r}$

Answer 1

Answer: (D)

If $n \geq r$, then ${ }^{n} C_{r-1}+{ }^{n} C_{r}={ }^{n+1} C_{r}$
Now, ${ }^{n} C_{r-1}+{ }^{n} C_{r}$
$=\frac{n !}{(n-r+1) !(r-1) !}+\frac{n !}{(n-r) ! r !}$
$=n !\left[\frac{r}{(n-r+1) ! r !}+\frac{n-r+1}{(n-r+1) ! r !}\right]$
$=n !\left[\frac{n+1}{(n-r+1) ! r !}\right]$
$={ }^{n+1} C_{r}$