| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Exactly N letters between items |
| Difficulty | Standard +0.3 Part (a) is a standard permutation with repeated letters (dividing by 3! for the three E's). Part (b) requires systematic case analysis (T...C vs C...T patterns) and careful counting of arrangements, which is moderately above routine but still a well-practiced technique in S1 permutations. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{9!}{3!}\) | M1 | \(\dfrac{9!}{e!}\), \(e = 2, 3\) |
| \(60480\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{7!}{3!} \times 2 \times 6\) | M1 | \(\dfrac{7!}{3!} \times k\) seen, \(k\) an integer \(> 0\) |
| M1 | \(\dfrac{m!}{n!} \times 2 \times q\) \(\;\) \(7 \leq m \leq 9\), \(1 \leq n \leq 3\), \(1 \leq q \leq 8\) all integers | |
| M1 | \(\dfrac{m!}{n!} \times p \times 6\) \(\;\) \(7 \leq m \leq 9\), \(1 \leq n \leq 3\), \(1 \leq p \leq 2\) all integers. (Accept 3P2 for 6). If M0 M0 M0 awarded, SC M1 for \(t \times 12\), \(t\) an integer \(\geq 20\), \(\dfrac{5!}{3!}\) | |
| \(10080\) | A1 | Exact value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{^7P_2 \times 6! \times 2}{3!}\) | M1 | \(\dfrac{6!}{3!} \times k\) seen, \(k\) an integer \(> 0\) |
| M1 | \(\dfrac{m!}{n!} \times\, ^7P_2 \times q\) \(\;\) \(m = 6,9\), \(1 \leq n \leq 3\), \(1 \leq q \leq 2\) all integers | |
| M1 | \(\dfrac{m!}{n!} \times\, ^7P_r \times 2\) \(\;\) \(m = 6,9\), \(1 \leq n \leq 3\), \(1 \leq r \leq 5\) all integers. If M0 M0 M0 awarded, SC M1 for \(t \times 84\), \(t\) an integer \(\geq 20\), \(\dfrac{5!}{3!}\) | |
| \(10080\) | A1 | Exact value |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{7!}{3!} \times 4P2\) | M1 | \(\frac{7!}{3!} \times k\) seen, \(k\) an integer \(> 0\) |
| M1 | \(t \times {}^4P_2\) or 12, \(t\) an integer \(\geqslant 20\), \(\frac{5!}{3!}\) | |
| M1 | \(\frac{m!}{n!} \times 4P2\), \(7 \leqslant m \leqslant 9\), \(1 \leqslant n \leqslant 3\) all integers | |
| 10008 | A1 | Exact value |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{9!}{3!}$ | M1 | $\dfrac{9!}{e!}$, $e = 2, 3$ |
| $60480$ | A1 | |
**Total: 2 marks**
---
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{7!}{3!} \times 2 \times 6$ | M1 | $\dfrac{7!}{3!} \times k$ seen, $k$ an integer $> 0$ |
| | M1 | $\dfrac{m!}{n!} \times 2 \times q$ $\;$ $7 \leq m \leq 9$, $1 \leq n \leq 3$, $1 \leq q \leq 8$ all integers |
| | M1 | $\dfrac{m!}{n!} \times p \times 6$ $\;$ $7 \leq m \leq 9$, $1 \leq n \leq 3$, $1 \leq p \leq 2$ all integers. (Accept 3P2 for 6). If **M0 M0 M0** awarded, **SC M1** for $t \times 12$, $t$ an integer $\geq 20$, $\dfrac{5!}{3!}$ |
| $10080$ | A1 | Exact value |
**Alternative method:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{^7P_2 \times 6! \times 2}{3!}$ | M1 | $\dfrac{6!}{3!} \times k$ seen, $k$ an integer $> 0$ |
| | M1 | $\dfrac{m!}{n!} \times\, ^7P_2 \times q$ $\;$ $m = 6,9$, $1 \leq n \leq 3$, $1 \leq q \leq 2$ all integers |
| | M1 | $\dfrac{m!}{n!} \times\, ^7P_r \times 2$ $\;$ $m = 6,9$, $1 \leq n \leq 3$, $1 \leq r \leq 5$ all integers. If **M0 M0 M0** awarded, **SC M1** for $t \times 84$, $t$ an integer $\geq 20$, $\dfrac{5!}{3!}$ |
| $10080$ | A1 | Exact value |
## Question 4(b) [Alternative method]:
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{7!}{3!} \times 4P2$ | M1 | $\frac{7!}{3!} \times k$ seen, $k$ an integer $> 0$ |
| | M1 | $t \times {}^4P_2$ or 12, $t$ an integer $\geqslant 20$, $\frac{5!}{3!}$ |
| | M1 | $\frac{m!}{n!} \times 4P2$, $7 \leqslant m \leqslant 9$, $1 \leqslant n \leqslant 3$ all integers |
| 10008 | A1 | Exact value |
---4
\begin{enumerate}[label=(\alph*)]
\item In how many different ways can the 9 letters of the word TELESCOPE be arranged?
\item In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q4 [6]}}