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

10 Professors and 20 Students
(a)2Pand 3S
$^{10} C_{2}$ x $^{20} C_{3}$ =45x(20x19x3) = 51300.
(b)Particular professor is included.
$^{9} C_{1}$ x $^{20} C_{3}$ = 9 X (20 X19 x 3) = 180 x 57 = 10260.
(c)Particular student included.
$^{10} C_{2}$ x $^{119} C_{2}$ = 45 x (19 x 9) = 7695.
(d)Particular Student excluded.
$^{10} C_{2}$ x $^{19} C_{3}$ =45 x (57x17) = 43605.

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10 Professors and 20 Students(a)2Pand 3S10C2 x 20C3 =45x(20x19x3) = 51300.(b)Particular professor is included.9C1 x 20C3 = 9 X (20 X19 x 3) = 180 x 57 = 10260.(c)Particular student included.10C2 x 119C2= 45 x (19 x 9) = 7695.(d)Particular Student excluded.10C2 x 19C3 =45 x (57x17) = 43605.