| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with positional constraints |
| Difficulty | Standard +0.8 This is a multi-part permutations question requiring careful handling of repeated letters, positional constraints, and conditional counting. Part (a) is routine (dividing by repeated letters), but parts (b) and (c) require systematic case-work: (b) needs fixing positions then subtracting arrangements with As together, while (c) demands enumeration of valid selections satisfying multiple conditions simultaneously. The combination of techniques and potential for counting errors places this above average difficulty. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{11!}{2!2!2!}\) | M1 | \(11!\) alone as numerator. \(2!\times m!\times n!\) on denominator, \(m=1,2\), \(n=1,2\). No additional terms, no additional operations |
| \(4989600\) | A1 | Exact answer only |
| Answer | Marks | Guidance |
|---|---|---|
| Arrange 7 letters CTEPILL = \(\frac{7!}{2!}\) | B1 | \(\frac{7!}{2!} \times k\) seen, \(k\) an integer \(> 1\) |
| Number of ways placing As in non-adjacent places \(= {}^8C_2\); \(\frac{7!}{2!} \times {}^8C_2\) | M1 | \(m \times n(n-1)\) or \(m \times {}^nC_2\) or \(m \times {}^nP_2\), \(n = 7, 8\) or \(9\), \(m\) an integer \(> 1\) |
| M1 | \(\frac{7!}{p!} \times {}^8C_2\) or \(\frac{7!}{p!} \times {}^8P_2\), \(p\) integer \(\geq 1\), condone \(2520 \times 28\) | |
| \(= 70560\) | A1 | Exact answer only. SC B1 70560 from M0, M1 only |
| Answer | Marks | Guidance |
|---|---|---|
| Total arrangements with R at beg. and end \(= \frac{9!}{2!2!}\) | M1 | \(\frac{9!}{2!m!} - k\), \(90720 > k\) integer \(> 1\), \(m = 1, 2\) |
| Arrangements with R at ends and As together \(= \frac{8!}{2!}\) | B1 | \(s - \frac{8!}{2!}\), \(s\) an integer \(> 1\) |
| With As not together \(= \frac{9!}{2!2!} - \frac{8!}{2!}\) | M1 | \(\frac{9!}{p} - \frac{8!}{q}\), \(p, q\) integers \(\geq 1\), condone \(90720 - 20160\) |
| \([90720 - 20160] = 70560\) | A1 | Exact answer only. SC B1 70560 from M0, M1 only |
| Answer | Marks | Guidance |
|---|---|---|
| \(RRAL\_\_\ {}^5C_2 = 10\); \(RRALL\_\ {}^5C_1 = 5\); \(RRAAL\_\ {}^5C_1 = 5\); \(RRAALL = 1\) | M1 | \({}^5C_x\) seen alone or \({}^5C_x \times k\), \(2 \geq k \geq 1\), \(k\) an integer, \(0 < x < 5\) linked to appropriate scenario |
| A1 | \({}^5C_2 \times k\), \(k=1\) oe or \({}^5C_1 \times m\), \(m = 1,2\) oe alone. SC if \({}^5C_x\) not seen. B2 for 5 or 10 linked to appropriate scenario | |
| M1 | Add outcomes from 3 or 4 identified correct scenarios only; \({}^2C_w \times {}^2C_x \times {}^2C_y \times {}^5C_z\), \(w+x+y+z=6\) identifies \(w\) Rs, \(\times\) As and \(y\) Ls | |
| \([\text{Total} =] 21\) | A1 | WWW, only dependent on 2nd M mark. Note: \({}^5C_2 + {}^5C_1 + {}^5C_1 + 1 = 21\) sufficient for 4/4. SC not all (or no) scenarios identified: B1 \(10+5+5+1\); DB1 \(= 21\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(RRAL\wedge\wedge = {}^7C_2\) | M1 | \({}^7C_x\) seen alone or \({}^7C_x \times k\), \(2 \geq k \geq 1\), \(k\) an integer, \(0 < x < 7\). Condone \({}^7P_x\) or \({}^7P_x \times k\) |
| M1 | \({}^7C_2 \times k\), \(2 \geq k \geq 1\) oe | |
| A1 | \({}^7C_2 \times k\), \(k = 1\) oe no other terms | |
| \([\text{Total} =] 21\) | A1 | Value stated |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{11!}{2!2!2!}$ | M1 | $11!$ alone as numerator. $2!\times m!\times n!$ on denominator, $m=1,2$, $n=1,2$. No additional terms, no additional operations |
| $4989600$ | A1 | Exact answer only |
## Question 6(b):
**Method 1:**
| Arrange 7 letters CTEPILL = $\frac{7!}{2!}$ | **B1** | $\frac{7!}{2!} \times k$ seen, $k$ an integer $> 1$ |
|---|---|---|
| Number of ways placing As in non-adjacent places $= {}^8C_2$; $\frac{7!}{2!} \times {}^8C_2$ | **M1** | $m \times n(n-1)$ or $m \times {}^nC_2$ or $m \times {}^nP_2$, $n = 7, 8$ or $9$, $m$ an integer $> 1$ |
| | **M1** | $\frac{7!}{p!} \times {}^8C_2$ or $\frac{7!}{p!} \times {}^8P_2$, $p$ integer $\geq 1$, condone $2520 \times 28$ |
| $= 70560$ | **A1** | Exact answer only. **SC B1** 70560 from M0, M1 only |
**Method 2:**
| Total arrangements with R at beg. and end $= \frac{9!}{2!2!}$ | **M1** | $\frac{9!}{2!m!} - k$, $90720 > k$ integer $> 1$, $m = 1, 2$ |
|---|---|---|
| Arrangements with R at ends and As together $= \frac{8!}{2!}$ | **B1** | $s - \frac{8!}{2!}$, $s$ an integer $> 1$ |
| With As not together $= \frac{9!}{2!2!} - \frac{8!}{2!}$ | **M1** | $\frac{9!}{p} - \frac{8!}{q}$, $p, q$ integers $\geq 1$, condone $90720 - 20160$ |
| $[90720 - 20160] = 70560$ | **A1** | Exact answer only. **SC B1** 70560 from M0, M1 only |
---
## Question 6(c):
**Method 1:**
| $RRAL\_\_\ {}^5C_2 = 10$; $RRALL\_\ {}^5C_1 = 5$; $RRAAL\_\ {}^5C_1 = 5$; $RRAALL = 1$ | **M1** | ${}^5C_x$ seen alone or ${}^5C_x \times k$, $2 \geq k \geq 1$, $k$ an integer, $0 < x < 5$ linked to appropriate scenario |
|---|---|---|
| | **A1** | ${}^5C_2 \times k$, $k=1$ oe or ${}^5C_1 \times m$, $m = 1,2$ oe alone. **SC** if ${}^5C_x$ not seen. **B2** for 5 or 10 linked to appropriate scenario |
| | **M1** | Add outcomes from 3 or 4 identified correct scenarios only; ${}^2C_w \times {}^2C_x \times {}^2C_y \times {}^5C_z$, $w+x+y+z=6$ identifies $w$ Rs, $\times$ As and $y$ Ls |
| $[\text{Total} =] 21$ | **A1** | WWW, only dependent on 2nd M mark. Note: ${}^5C_2 + {}^5C_1 + {}^5C_1 + 1 = 21$ sufficient for 4/4. **SC** not all (or no) scenarios identified: **B1** $10+5+5+1$; **DB1** $= 21$ |
**Method 2:**
| $RRAL\wedge\wedge = {}^7C_2$ | **M1** | ${}^7C_x$ seen alone or ${}^7C_x \times k$, $2 \geq k \geq 1$, $k$ an integer, $0 < x < 7$. Condone ${}^7P_x$ or ${}^7P_x \times k$ |
|---|---|---|
| | **M1** | ${}^7C_2 \times k$, $2 \geq k \geq 1$ oe |
| | **A1** | ${}^7C_2 \times k$, $k = 1$ oe no other terms |
| $[\text{Total} =] 21$ | **A1** | Value stated |
---6
\begin{enumerate}[label=(\alph*)]
\item Find the total number of different arrangements of the 11 letters in the word CATERPILLAR.
\item Find the total number of different arrangements of the 11 letters in the word CATERPILLAR in which there is an R at the beginning and an R at the end, and the two As are not together. [4]
\item Find the total number of different selections of 6 letters from the 11 letters of the word CATERPILLAR that contain both Rs and at least one A and at least one L.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q6 [10]}}