Permutations & Combinations: worked solution — RVHS 2025 H2 Math Prelim Paper 2
River Valley High School2025 PrelimPaper 2●●○ Standard10 marks
Question
A Pizzeria sells pizza by individual slices. 8 slices forms a full pizza.
- A group of three friends bought 4 slices of Hawaiian, 1 slice of pepperoni, 1 slice of seafood, 1 slice of cheese, and 1 slice of vegetarian pizzas. Find the number of ways the pizza slices bought could be arranged in a circle such that the 4 slices of Hawaiian pizzas are placed together.
- The three friends each eat a slice of Hawaiian pizza. They then want to distribute the remaining 5 slices amongst the three of them. Find the number of ways this can be done such that everyone gets at least one more slice of pizza.
Show full worked solution▾
(a) Number of ways \(= (5-1)! = 24\)
(b)
Case 1: One person gets 3 more slices \[ \binom{3}{1}\times\frac{5!}{3!} = 60 \]
Case 2: Two people get 2 more slices \[ \binom{3}{2}\times\frac{5!}{2!\,2!} = 90 \]
Total number of ways \(= 150\)
Alternatively:
Case 1: \(\displaystyle\binom{3}{1}\binom{5}{3}\binom{2}{1} = 60\)
Case 2: \(\displaystyle\binom{3}{2}\binom{5}{2}\binom{3}{2} = 90\)
Answer: (a) 24 (b) 150