Question

There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is be formed. Find the number of ways in which this can be done. Further find in how many of these committees :

(i) A particular professor is included.

(ii) A particular student is included.

(iii) A particular student is excluded

Answer 1

Total number of professor = 10

Total number of student = 20

Committee of 2 professor and 3 student can be selected in ${}^{10}{C}_2{\times }^{20}{C}_3$ ways.

$=\frac{10!}{2!8!}\times \frac{20!}{3!17!}$

$=\frac{10\times 9}{2}\times \frac{20\times 19\times 18}{3\times 2}$

(i) A particular professor is included

$\therefore$ committee is ${}^9{C}_1{\times }^{20}{C}_3$

$=\frac{9!}{8!}\times \frac{20}{3!\times 17!}=\frac{9\times 20\times 19\times 18}{3\times 2}$

$=10260$

(ii) A particular student is included

$\therefore$ Committee is ${}^{10}{C}_2{\times }^{19}{C}_2$

$=\frac{10!}{2!\times 8!}\times \frac{19}{2!\times 17!}=\frac{10\times 9\times 19\times 18}{2\times 2\times 1}=7695$

(iii) A particular student is excluded $\to$ now total students are 19

$\therefore$ Committee is ${}^{10}{C}_2{\times }^{19}{C}_3$

$=\frac{10!}{2!\times 8!}\times \frac{19}{3!\times 16!}=\frac{10\times 9\times 19\times 18\times 17}{2\times 3\times 2}$

$=43605$