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. ${}^5{C}_2{\times }^{22}{C}_{13}$
• B. ${}^5{C}_3{\times }^{22}{C}_9$
• C. ${}^5{C}_3{\times }^{23}{C}_8$
• D. ${}^8{C}_5{\times }^{21}{C}_7$

Answer 1

Answer: (A), (B)

The correct options are
A ${}^5{C}_2{\times }^{22}{C}_{13}$
B ${}^5{C}_3{\times }^{22}{C}_9$

$3$ vacancies for SC candidates can be filled up from $5$ candidates in ${}^5{C}_3$ 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

${}^5{C}_3\times {}^{22}{C}_9$
$={}^5{C}_3\times {}^{22}{C}_9={}^5{C}_{5-3}{\times }^{22}{C}_{22-9}={}^5{C}_2\times {}^{22}{C}_{13}$