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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

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Solution

## Total number of professor = 10 Total number of student = 20 Committee of 2 professor and 3 student can be selected in 10C2×20C3 ways. =10!2!8!×20!3!17! =10×92×20×19×183×2 (i) A particular professor is included ∴ committee is 9C1×20C3 =9!8!×203!×17!=9×20×19×183×2 =10260 (ii) A particular student is included ∴ Committee is 10C2×19C2 =10!2!×8!×192!×17!=10×9×19×182×2×1=7695 (iii) A particular student is excluded → now total students are 19 ∴ Committee is 10C2×19C3 =10!2!×8!×193!×16!=10×9×19×18×172×3×2 =43605

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