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?

Number of ways in which committee can be formed, is

^{10} C_{2} \times^{20} C_{3}

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Number of ways in which a particular professor is included, is

^{9} C_{1} \times^{20} C_{3}

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

^{10} C_{2} \times^{19} C_{2}

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

^{10} C_{2} \times^{19} C_{3}

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Solution

The correct options are
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
$^{10} C_{2} \times^{19} C_{3}$
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
$^{10} C_{2} \times^{20 - 1} C_{3} =^{10} C_{2} \times^{19} C_{3}$

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