Question

A primary school has 10 students, and we’d like to award 3 prizes to 3 different students. Each student can receive at most one prize. How many ways are there to award the prizes if:

(1) All the prizes are the same.

(2) All the prizes are the same and Tom, the youngest student, is awarded a prize.

(3 )All the prizes are different.

(4) All the prizes are different and Tom, the youngest student is not awarded a prize.

Answer 1

1) If all the prizes are same them arrangements doesn't matter, only selection of 3 students matters. So

no of ways = n(selection of 3 students from 10 students)

= C(10,3) = 10!/(7!3!) = 120

2) If Tom has awarded a prize then we only have to select 2 more students from the remaining 9 students.

No of ways = C(9,2) = 9!/(7!2!) = 36

3) Since all prizes are different so arrangements also matters. No. of ways =Permutation of 3 things among 10

Required ways = P(10,3) = 10!/7! = 720

4) Since youngest one is not getting any prize. So we have to choose and arrange 3 students from 9 remaining.

Required no of ways = P(9,3) = 9!/6! = 504