A) \[^{n}{{C}_{r-1}}\]
B) \[^{n}{{C}_{r}}\]
C) \[^{n-1}{{C}_{r}}\]
D) \[^{n+1}{{C}_{r}}\]
Correct Answer: D
Solution :
Key Idea: If\[n\ge r\], then \[^{n}{{C}_{r-1}}{{+}^{n}}{{C}_{r}}{{=}^{n+1}}{{C}_{r}}\] Now, \[^{n}{{C}_{r-1}}{{+}^{n}}{{C}_{r}}\] \[=\frac{n!}{(n-r+1)!(r-1)!}+\frac{n!}{(n-r)!r!}\] \[=n!\left[ \frac{r}{(n-r+1)!r!}+\frac{n-r+1}{(n-r+1)!r!} \right]\] \[=n!\left[ \frac{n+1}{(n-r+1)!r!} \right]\] \[{{=}^{n+1}}{{C}_{r}}\] Note: \[^{n}{{C}_{r}}\] cannot defined if\[n<r\].
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