Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 A café has 4 entrées, 5 mains and 3 desserts. A meal deal is either an entrée and a main, or a main and a dessert. How many different meal deals are there?
Show worked solution
Entrée and main: 4 × 5 = 20. Main and dessert: 5 × 3 = 15. These are separate cases, so add: 20 + 15 = 35. 60 = 4 × 5 × 3 is a three-course meal; 12 = 4 + 5 + 3 adds the dishes; 20 counts only the first kind.
2 Codes of 3 letters are made from A, B, C, D, E. How many codes are there (a) if letters may repeat, and (b) if no letter repeats and the code must start with a vowel?
Show worked solution
(a) 5 × 5 × 5 = 125. (b) The first letter is A or E: 2 ways. Then 4 letters are left, then 3: 2 × 4 × 3 = 24.
3 Using the digits below, each at most once, how many odd 3-digit numbers can be made?
Show worked solution
Units first: 1, 3 or 5, so 3 ways. Hundreds: any of the 4 digits left. Tens: any of the 3 left. 3 × 4 × 3 = 36.
Level 2 · Core
4 4 boys and 3 girls stand in a row. The three girls must stand together. How many arrangements are possible?
Show worked solution
4 boys + 1 girl-block = 5 items, arranged in 5! = 120 ways. Inside the block the girls go in 3! = 6 orders. 120 × 6 = 720. 120 forgets the order inside the block; 144 = 4! × 3! keeps the block fixed at one end; 5040 = 7! ignores the condition.
5 A team of 4 is chosen from 6 boys and 5 girls. The team must have exactly 2 girls.
Show worked solution
Girls: 5 × 4 ÷ 2! = 10. Boys: 6 × 5 ÷ 2! = 15. Each pair of girls goes with each pair of boys: 10 × 15 = 150 teams.
6 In how many different orders can the letters of COFFEE be arranged?
Show worked solution
Six letters give 6! = 720 orders if all were different. F appears twice and E appears twice, so divide by 2! × 2! = 4: 720 ÷ 4 = 180.
Level 3 · Challenge
7 Three statements:
1 Six people can sit around a round table in 6! = 720 different ways.
2 Choosing 3 people from 8 can be done in as many ways as choosing 5 people from 8.
3 The letters of LETTERS can be arranged in 1260 ways.
Which statements are correct?
1 Six people can sit around a round table in 6! = 720 different ways.
2 Choosing 3 people from 8 can be done in as many ways as choosing 5 people from 8.
3 The letters of LETTERS can be arranged in 1260 ways.
Which statements are correct?
Show worked solution
Rotating a seating does not change it, so 6 people give 6! ÷ 6 = 120 seatings, not 720 (1 ✗). Choosing 3 to be in is the same as choosing 5 to leave out: both are 8 × 7 × 6 ÷ 3! = 56 (2 ✓). LETTERS has 7 letters with E twice and T twice: 5040 ÷ 4 = 1260 (3 ✓). 2 and 3.
8 A PIN has 4 digits, each from 0 to 9 (0 may come first).
Show worked solution
All PINs: 104 = 10 000. All different: 10 × 9 × 8 × 7 = 5040. At least one repeat: 10 000 − 5040 = 4960.
9 These 8 people split into two teams of 4. The teams have no names. In how many ways can this be done?
Show worked solution
Choosing 4 people: 8 × 7 × 6 × 5 ÷ 4! = 70. But the split ABCD | EFGH is counted twice, once when ABCD is chosen and once when EFGH is chosen, and the teams have no names. So 70 ÷ 2 = 35. Check: A's three teammates come from the other 7: 7 × 6 × 5 ÷ 3! = 35.