Permutations And Combinations Question 160

Question: The value of $ \sum\limits _{r=0}^{n-1}{\frac{^{n}C _{r}}{^{n}C _{r}+{{}^{n}}{C _{r+1}}}} $ equals [MP PET 2004]

Options:

A) $ n+1 $

B) $ \frac{n}{2} $

C) $ n+2 $

D) None of these

Show Answer

Answer:

Correct Answer: B

Solution:

  • $ \sum\limits _{r=0}^{n-1}{\frac{^{n}C _{r}}{^{n}C _{r}+{{}^{n}}{C _{r+1}}}}=\sum\limits _{r=0}^{n-1}{\frac{1}{1+\frac{^{n}{C _{r+1}}}{^{n}C _{r}}}}=\sum\limits _{r=0}^{n-1}{\frac{1}{1+\frac{n-r}{r+1}}} $ $ =\sum\limits _{r=0}^{n-1}{\frac{r+1}{n+1}}=\frac{1}{n+1}\sum\limits _{r=0}^{n-1}{(r+1)} $ $ =\frac{1}{(n+1)}[1+2+…+n]=\frac{n}{2} $ .

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