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Question

To fill up $$12$$ vacancies, there are $$25$$ candidates of which $$5$$ are from SC. If $$3$$ of these vacancies are reserved for SC candidates while the remaining are open to all then the number of ways in which the selection can be made is :


A
5C2×22C13
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B
5C3×22C9
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C
5C3×23C8
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D
8C5×21C7
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Solution

The correct options are
A $$^5C_2\times ^{22}C_{13}$$
B $$^5C_3\times ^{22}C_{9}$$
$$3$$ vacancies for SC candidates can be filled up from $$5$$ candidates in $$^5C_3$$ ways.
After this for remaining $$12-3=9$$ vacancies, there will be $$25-3=22$$ candidates. 
These vacancies can be filled up in $$^{22}C_9$$ ways. 
Hence, the required number of ways are 
$$^5C_3\times {}^{22}C_9$$
$$={}^5C_3\times {}^{22}C_9={}^5C_{5-3}\times ^{22}C_{22-9}={}^5C_2\times {}^{22}C_{13}$$

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