Answer:

Correct Answer: A

Solution:

• [a] The required number of ways $ {{=}^{21}}C _{10}{{+}^{22}}C _{10}{{+}^{23}}C _{10}+…{{+}^{30}}C _{10} $ $ ={{(}^{21}}C _{10}{{+}^{21}}C _{11}{{+}^{22}}C _{12}{{+}^{23}}C _{13}+…{{+}^{30}}C _{20}){{-}^{21}}C _{11} $ $ {{=}^{22}}C _{11}{{+}^{22}}C _{12}{{+}^{23}}C _{13}+…{{+}^{30}}C _{20}{{-}^{21}}C _{10} $ $ ({{\because }^{n}}{C _{r-1}}{{+}^{n}}C _{r}{{=}^{n+1}}C _{r}) $ $ {{=}^{31}}C _{20}{{-}^{21}}C _{10}. $