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Question

# There are 10 professors and 20 students out of whom a committee of 2 professors and 3 students is to 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.

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Solution

## Clearly, 2 professors and 3 students are selected out of 10 professors and 20 students, respectively. Required number of ways = ${}^{10}{C}_{2}{×}^{20}{C}_{3}=\frac{10}{2}×\frac{9}{1}×\frac{20}{3}×\frac{19}{2}×\frac{18}{1}=51300$ (i) If a particular professor is included, it means that 1 professor is selected out of the remaining 9 professors. Required number of ways = ${}^{20}{C}_{3}{×}^{9}{C}_{1}=\frac{20}{3}×\frac{19}{2}×\frac{18}{1}×\frac{9}{1}=10260$ (ii) If a particular student is included, it means that 2 students are selected out of the remaining 19 students. Required number of ways = ${}^{19}{C}_{2}{×}^{10}{C}_{2}=\frac{19}{2}×\frac{18}{1}×\frac{10}{2}×\frac{9}{1}=7695$ (iii) If a particular student is excluded, it means that 3 students are selected out of the remaining 19 students. Required number of ways = ${}^{19}{C}_{3}{×}^{10}{C}_{2}=\frac{19}{3}×\frac{18}{2}×\frac{17}{1}×\frac{10}{2}×\frac{9}{1}=43605$

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