Question -
Answer -
Given:
Total number ofprofessor = 10
Total number ofstudents = 20
Number of ways =(choosing 2 professors out of 10 professors) × (choosing 3 students out of 20students)
= (10C2)× (20C3)
By using the formula,
nCr = n!/r!(n – r)!
10C2 × 20C3 =10!/2!(10 – 2)! × 20!/3!(20-3)!
= 10!/(2! 8!) ×20!/(3! 17!)
= [10×9×8!]/(2! 8!) ×[20×19×18×17!]/(17! 3!)
= [10×9]/2! ×[20×19×18]/(3!)
= [10×9]/(2×1) ×[20×19×18]/(3×2×1)
= 5×9 × 10×19×6
= 45 × 1140
= 51300 ways
(i) a particular professoris included.
Number of ways =(choosing 1 professor out of 9 professors) × (choosing 3 students out of 20students)
= 9C1 × 20C3
By using the formula,
nCr = n!/r!(n – r)!
9C1 × 20C3 =9!/1!(9 – 1)! × 20!/3!(20-3)!
= 9!/(1! 8!) × 20!/(3!17!)
= [9×8!]/(8!) ×[20×19×18×17!]/(17! 3!)
= 9 × [20×19×18]/(3!)
= 9×[20×19×18]/(3×2×1)
= 9 × 10×19×6
= 10260 ways
(ii) a particular studentis included.
Number of ways =(choosing 2 professors out of 10 professors) × (choosing 2 students out of 19students)
= 10C2 × 19C2
By using the formula,
nCr = n!/r!(n – r)!
10C2 × 19C2 =10!/2!(10 – 2)! × 19!/2!(19-2)!
= 10!/(2! 8!) ×19!/(2! 17!)
= [10×9×8!]/(2! 8!) ×[19×18×17!]/(17! 2!)
= [10×9]/2! ×[19×18]/(2!)
= [10×9]/(2×1) ×[19×18]/(2×1)
= 5×9 × 19×9
= 45 × 171
= 7695 ways
(iii) a particular studentis excluded.
Number of ways =(choosing 2 professors out of 10 professors) × (choosing 3 students out of 19students)
= 10C2 × 19C3
By using the formula,
nCr = n!/r!(n – r)!
10C2 × 19C3 =10!/2!(10 – 2)! × 19!/3!(19-3)!
= 10!/(2! 8!) ×19!/(3! 16!)
= [10×9×8!]/(2! 8!) ×[19×18×17×16!]/(16! 3!)
= [10×9]/2! ×[19×18×17]/(3!)
= [10×9]/(2×1) ×[19×18×17]/(3×2×1)
= 5×9 × 19×3×17
= 45 × 969
= 43605 ways
∴ The required no.of ways are 51300, 10260, 7695, 43605.