| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with identical objects |
| Difficulty | Moderate -0.8 This is a straightforward permutations question requiring basic counting principles. Part (i) uses the standard formula for arrangements with identical objects (13!/(9!4!)), part (ii) treats grouped spaces as a single unit (10 arrangements), and part (iii) applies simple probability. All techniques are routine for S1 level with no novel problem-solving required. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| For using a permutation involving 13 | M1 | |
| For correct answer | A1 | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| For using a 10 | M1 | |
| For using a 9! | M1 | |
| For correct answer | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| For a subtraction of a suitable prob \(< 1\), from 1 | M1 | |
| For correct answer, ft on their (i) and (ii) | A1 ft | 2 marks |
**(i)** $_13P_9 = 259,459,200$ or $259,000,000$
| For using a permutation involving 13 | M1
| For correct answer | A1 | 2 marks
**(ii)** $10!$ or $_10p = 3628800$
| For using a 10 | M1
| For using a 9! | M1
| For correct answer | A1 | 3 marks
**(iii)** $1 - (ii) / (i) = 0.986$
| For a subtraction of a suitable prob $< 1$, from 1 | M1
| For correct answer, ft on their (i) and (ii) | A1 ft | 2 marks3 A staff car park at a school has 13 parking spaces in a row. There are 9 cars to be parked.\\
(i) How many different arrangements are there for parking the 9 cars and leaving 4 empty spaces?\\
(ii) How many different arrangements are there if the 4 empty spaces are next to each other?\\
(iii) If the parking is random, find the probability that there will not be 4 empty spaces next to each other.
\hfill \mbox{\textit{CAIE S1 2005 Q3 [7]}}