| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with positional constraints |
| Difficulty | Moderate -0.8 This is a straightforward permutations question with repeated elements. Part (i) uses the standard formula n!/r! for arrangements with repetition (6!/3! = 120). Part (ii) requires simple case analysis (odd-odd positions) but involves only basic counting principles with no complex constraints or novel problem-solving. Below average difficulty for A-level, as it's a direct application of textbook formulas. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{6!}{3!} = 120\) | M1, A1 [2] | For dividing by \(3!\); Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(5\ldots7 = \frac{4!}{2!} = 12\) | M1 | For identifying different cases |
| B1 | For \(4!/2!\) seen | |
| \(7\ldots5 = \frac{4!}{2!} = 12\) | B1 | For \(4!\) alone seen or in a sum or product |
| \(7\ldots7 = 4! = 24\) | ||
| \(\text{total} = 48\) | A1 [4] | Correct final answer |
## Question 3:
**Part (i)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{6!}{3!} = 120$ | M1, A1 **[2]** | For dividing by $3!$; Correct answer |
**Part (ii)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $5\ldots7 = \frac{4!}{2!} = 12$ | M1 | For identifying different cases |
| | B1 | For $4!/2!$ seen |
| $7\ldots5 = \frac{4!}{2!} = 12$ | B1 | For $4!$ alone seen or in a sum or product |
| $7\ldots7 = 4! = 24$ | | |
| $\text{total} = 48$ | A1 **[4]** | Correct final answer |
---3 The six digits 4, 5, 6, 7, 7, 7 can be arranged to give many different 6-digit numbers.\\
(i) How many different 6-digit numbers can be made?\\
(ii) How many of these 6-digit numbers start with an odd digit and end with an odd digit?
\hfill \mbox{\textit{CAIE S1 2007 Q3 [6]}}