Question

There are $10$ professors and $20$ lecturers, out of whom a committee of $2$ professors and $3$ lecturers is to be formed. Which of the following options is (are) CORRECT?

• A. Number of ways in which committee can be formed, is ${}^{10}{C}_2\times {}^{20}{C}_3$
• B. Number of ways in which a particular professor is included, is ${}^9{C}_1\times {}^{20}{C}_3$
  
• C. Number of ways in which a particular lecturer is included, is
  ${}^{10}{C}_2\times {}^{19}{C}_2$
• D. Number of ways in which a particular lecturer is excluded, is
  ${}^{10}{C}_2\times {}^{19}{C}_3$

Answer 1

Answer: (A), (B), (C), (D)

Number of ways in which a particular lecturer is excluded, is

${}^{10}{C}_2\times {}^{19}{C}_3$
Number of ways of selecting $2$ professors out of $10$ and $3$ lecturers out of $20$ is ${}^{10}{C}_2\times {}^{20}{C}_3$

Number of ways when a particular professor is included, is
${}^{10-1}{C}_1\times {}^{20}{C}_3={}^9{C}_1\times {}^{20}{C}_3$

Number of ways when particular lecturer is included, is ${}^{10}{C}_2\times {}^{19}{C}_2$

Number of ways when a particular lecturer is excluded, is
${}^{10}{C}_2\times {}^{20-1}{C}_3={}^{10}{C}_2\times {}^{19}{C}_3$