Question -
Answer -
Given:
Total number ofprofessor = 10
Total number ofstudents = 20
Number of ways =(choosing 2 professors out of 10 professors) ├Ч (choosing 3 students out of 20students)
= (10C2)├Ч (20C3)
By using the formula,
nCr┬а= n!/r!(n тАУ r)!
10C2┬а├Ч┬а20C3┬а=10!/2!(10 тАУ 2)! ├Ч 20!/3!(20-3)!
= 10!/(2! 8!) ├Ч20!/(3! 17!)
= [10├Ч9├Ч8!]/(2! 8!) ├Ч[20├Ч19├Ч18├Ч17!]/(17! 3!)
= [10├Ч9]/2! ├Ч[20├Ч19├Ч18]/(3!)
= [10├Ч9]/(2├Ч1) ├Ч[20├Ч19├Ч18]/(3├Ч2├Ч1)
= 5├Ч9 ├Ч 10├Ч19├Ч6
= 45 ├Ч 1140
= 51300 ways
(i)┬аa particular professoris included.
Number of ways =(choosing 1 professor out of 9 professors) ├Ч (choosing 3 students out of 20students)
=┬а9C1┬а├Ч┬а20C3
By using the formula,
nCr┬а= n!/r!(n тАУ r)!
9C1┬а├Ч┬а20C3┬а=9!/1!(9 тАУ 1)! ├Ч 20!/3!(20-3)!
= 9!/(1! 8!) ├Ч 20!/(3!17!)
= [9├Ч8!]/(8!) ├Ч[20├Ч19├Ч18├Ч17!]/(17! 3!)
= 9 ├Ч [20├Ч19├Ч18]/(3!)
= 9├Ч[20├Ч19├Ч18]/(3├Ч2├Ч1)
= 9 ├Ч 10├Ч19├Ч6
= 10260 ways
(ii)┬аa particular studentis included.
Number of ways =(choosing 2 professors out of 10 professors) ├Ч (choosing 2 students out of 19students)
=┬а10C2┬а├Ч┬а19C2
By using the formula,
nCr┬а= n!/r!(n тАУ r)!
10C2┬а├Ч┬а19C2┬а=10!/2!(10 тАУ 2)! ├Ч 19!/2!(19-2)!
= 10!/(2! 8!) ├Ч19!/(2! 17!)
= [10├Ч9├Ч8!]/(2! 8!) ├Ч[19├Ч18├Ч17!]/(17! 2!)
= [10├Ч9]/2! ├Ч[19├Ч18]/(2!)
= [10├Ч9]/(2├Ч1) ├Ч[19├Ч18]/(2├Ч1)
= 5├Ч9 ├Ч 19├Ч9
= 45 ├Ч 171
= 7695 ways
(iii)┬аa particular studentis excluded.
Number of ways =(choosing 2 professors out of 10 professors) ├Ч (choosing 3 students out of 19students)
=┬а10C2┬а├Ч┬а19C3
By using the formula,
nCr┬а= n!/r!(n тАУ r)!
10C2┬а├Ч┬а19C3┬а=10!/2!(10 тАУ 2)! ├Ч 19!/3!(19-3)!
= 10!/(2! 8!) ├Ч19!/(3! 16!)
= [10├Ч9├Ч8!]/(2! 8!) ├Ч[19├Ч18├Ч17├Ч16!]/(16! 3!)
= [10├Ч9]/2! ├Ч[19├Ч18├Ч17]/(3!)
= [10├Ч9]/(2├Ч1) ├Ч[19├Ч18├Ч17]/(3├Ч2├Ч1)
= 5├Ч9 ├Ч 19├Ч3├Ч17
= 45 ├Ч 969
= 43605 ways
тИ┤┬аThe required no.of ways are 51300, 10260, 7695, 43605.