A) \[^{n+4}{{C}_{r}}\]
B) \[{{2.}^{n+4}}{{C}_{r-1}}\]
C) \[{{4.}^{n}}{{C}_{r}}\]
D) \[{{11.}^{n}}{{C}_{r}}\]
Correct Answer: A
Solution :
[a] \[^{n}{{C}_{7}}+{{4.}^{n}}{{C}_{r-1}}+{{6.}^{n}}{{C}_{r-2}}+{{4.}^{n}}{{C}_{r-3}}{{+}^{n}}{{C}_{r-4}}\] |
\[={{(}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}})+3{{(}^{n}}{{C}_{r-1}}{{+}^{n}}{{C}_{r-2}})\] |
\[+3{{(}^{n}}{{C}_{r-2}}{{+}^{n}}{{C}_{r-3}})+{{(}^{n}}{{C}_{r-3}}{{+}^{n}}{{C}_{r-4}})\] |
\[=n{{+}^{1}}{{C}_{r}}+{{3.}^{n+1}}{{C}_{r-1}}+{{3.}^{n+1}}{{C}_{r-2}}{{+}^{n+1}}{{C}_{r-3}}\] |
\[={{(}^{n+1}}{{C}_{r}}{{+}^{n+1}}{{C}_{r-1}})+2{{(}^{n+1}}{{C}_{r-1}}+\] |
\[+{{(}^{n+1}}{{C}_{r-2}}{{+}^{n+1}}{{C}_{r-3}})\] |
\[{{=}^{n+2}}{{C}_{r}}+2.{{(}^{n+2}}{{C}_{r-1}}{{+}^{n+2}}{{C}_{r-2}})\] |
\[={{(}^{n+2}}{{C}_{r}}{{+}^{n+2}}{{C}_{r-1}})+{{(}^{n+2}}{{C}_{r-1}}{{+}^{n+2}}{{C}_{r-2}})\] |
\[{{=}^{n+3}}{{C}_{r}}{{+}^{n+3}}{{C}_{r-1}}{{=}^{n+4}}{{C}_{r}}\] |
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