| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with adjacency requirements |
| Difficulty | Moderate -0.3 This is a standard permutations question testing arrangements with repeated letters and basic selection. Part (i)(a) is direct application of n!/n₁!n₂!... formula, (i)(b) requires the standard 'treat as one unit' technique, and (ii) combines selection with restrictions. All techniques are routine for S1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{12!}{4!\,2!} = 9979200\) (9980000) | B1 | Dividing by \(4!\) and \(2!\) only |
| B1 2 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{9!}{2!} = 181440\) (181000) | B1 | \(9!\) or \(9 \times 8!\) seen not in denom |
| B1 2 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(_{6}C_2\) or \(_{4}C_0 \times {}_{2}C_2 \times {}_{6}C_2\) or \(_{6}C_4\) or \(_{6}P_2/2!\) | M1 | For seeing \({}_{6}C_{\text{something}}\) or \({}_{6}P_{\text{something}}\) in a product (could be with 1) |
| M1 | For seeing \({}_{\text{something}}C_2\) or \({}_{6}C_4\) | |
| \(= 15\) | A1 3 | Correct answer; 15 with no working scores full marks |
## Question 5:
### Part (i)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{12!}{4!\,2!} = 9979200$ (9980000) | B1 | Dividing by $4!$ and $2!$ only |
| | B1 **2** | Correct answer |
### Part (i)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{9!}{2!} = 181440$ (181000) | B1 | $9!$ or $9 \times 8!$ seen not in denom |
| | B1 **2** | Correct answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $_{6}C_2$ or $_{4}C_0 \times {}_{2}C_2 \times {}_{6}C_2$ or $_{6}C_4$ or $_{6}P_2/2!$ | M1 | For seeing ${}_{6}C_{\text{something}}$ or ${}_{6}P_{\text{something}}$ in a product (could be with 1) |
| | M1 | For seeing ${}_{\text{something}}C_2$ or ${}_{6}C_4$ |
| $= 15$ | A1 **3** | Correct answer; 15 with no working scores full marks |
---5 (i) Find the number of ways in which all twelve letters of the word REFRIGERATOR can be arranged
\begin{enumerate}[label=(\alph*)]
\item if there are no restrictions,
\item if the Rs must all be together.\\
(ii) How many different selections of four letters from the twelve letters of the word REFRIGERATOR contain no Rs and two Es?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2007 Q5 [7]}}