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:

# 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:

1. A

${2}^{9}$

2. B

${2}^{10}$

3. C

${2}^{11}$

4. D

${2}^{11}{-}^{11}{C}_{5}$

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### Solution:

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 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

$2S={2}^{11}⇒S={2}^{10}$

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