In an examination, there are 11 papers. A candidate has to pass in at least 6 papers to pass the examination. The number of ways in which the candidate can pass the examination is:

Number of ways passing the examination is

$S =^{11} C_{6} +^{11} C_{7} +^{11} C_{8} +^{11} C_{9} +^{11} C_{10} +^{11} C_{11}$ (1)

Using $^{n} C_{r} =^{n} C_{n - r}$ we get

$S =^{11} C_{5} +^{11} C_{4} +^{11} C_{3} +^{11} C_{2} +^{11} C_{1} +^{11} C_{0}$ (2)

Adding (1) and (2) we get

$2 S = 2^{11} \Rightarrow S = 2^{10}$

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