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

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