in how many ways can 4 prizes be distributed among 5 person when
(i) no person get more than 1.
Choose the 1 person to get no prize in 5 ways.
Answer: 5 ways
(ii) a person may get any number of prizes.
The cases are the ways to split 4 prizes,
1. 1+1+1+1
2. 2+1+1
3. 2+2
4. 3+1
5. 4
Case 1: no person gets more than 1 prize. Same as (i), 5 ways
Case 2: 1 person gets 2 prizes, 2 other people get 1 prize each.
Choose the person to get 2 prizes 5 ways.
Choose two people to get 1 prize each from the remaining 4 in C(4,2) = 6 ways.
That's 5×6 or 30 ways.
Case 3: 2 people get 2 prizes each.
Choose those 2 any of C(5,2) = 10 ways.
Case 4: 1 person gets 3 prizes, 1 other person gets 1 prize.
Choose the person to get 3 prizes 5 ways.
Choose the person to get 1 prize 4 ways.
That's 5×4 = 20 ways.
Case 5: 1 person gets all 4 prizes.
Choose that person 5 ways.
Answer: 5+30+10+20+5 = 70 ways
(iii) a person get all the prizes.
That's case 5 of (ii).
Answer: 5 ways.
Edwin