| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with grouped categories |
| Difficulty | Moderate -0.3 Part (a) is a standard grouped arrangement problem with repeated letters requiring treating groups as single units then accounting for internal arrangements. Part (b) uses complementary counting. Both are routine S1 permutations techniques with straightforward application, making this slightly easier than average but still requiring careful systematic work with the repeated letters. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(5!\) | M1 | \(k!\) where \(k = 5, 6\) or \(7\); Condone \(\times 1\) OE |
| \(120\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Total no. of ways \(= \dfrac{8!}{2!3!} = 3360\) | M1 | \(\dfrac{8!}{a!b!}\), \(a=1,2\); \(b=1,3\); \(a \neq b\) |
| With 3Es together \(= \dfrac{6!}{2!} = 360\) | M1 | \(\dfrac{6!}{c!}\), \(c=1,2\) seen in an addition/subtraction |
| With 3Es not together \(= 3360 - 360\) | M1 | \(\dfrac{8!}{d!e!} - \dfrac{6!}{f!}\) where \(d,f=1,2\) & \(e=1,3\) |
| \(3000\) | A1 | |
| Total: 4 |
## Question 1:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $5!$ | M1 | $k!$ where $k = 5, 6$ or $7$; Condone $\times 1$ OE |
| $120$ | A1 | |
| **Total: 2** | | |
---
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Total no. of ways $= \dfrac{8!}{2!3!} = 3360$ | M1 | $\dfrac{8!}{a!b!}$, $a=1,2$; $b=1,3$; $a \neq b$ |
| With 3Es together $= \dfrac{6!}{2!} = 360$ | M1 | $\dfrac{6!}{c!}$, $c=1,2$ seen in an addition/subtraction |
| With 3Es not together $= 3360 - 360$ | M1 | $\dfrac{8!}{d!e!} - \dfrac{6!}{f!}$ where $d,f=1,2$ & $e=1,3$ |
| $3000$ | A1 | |
| **Total: 4** | | |1
\begin{enumerate}[label=(\alph*)]
\item Find the number of different arrangements of the 8 letters in the word DECEIVED in which all three Es are together and the two Ds are together.
\item Find the number of different arrangements of the 8 letters in the word DECEIVED in which the three Es are not all together.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q1 [6]}}