| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Standard +0.3 This is a standard combinations and permutations question with multiple parts requiring systematic case-by-case analysis. Part (a) involves straightforward application of combination formulas with constraints, part (b) uses the standard 'treat groups as units' technique, and part (c) requires the gap method for non-adjacent arrangements. While multi-step, all techniques are textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(S+4C+2R\): \({}^6C_1 \times {}^8C_4 \times {}^{11}C_2 [= 6 \times 70 \times 55] = 23100\) | M1 | \({}^6C_e \times {}^8C_f \times {}^{11}C_g\), with \(e+f+g=7\) seen |
| \(S+5C+1R\): \({}^6C_1 \times {}^8C_5 \times {}^{11}C_1 [= 6 \times 56 \times 11] = 3696\) | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified, www |
| \(S+6C[+0R]\): \({}^6C_1 \times {}^8C_6 \times [{}^{11}C_0][= 6 \times 28] = 168\) | M1 | Add values of 3 correct scenarios. No incorrect scenarios, no repeated scenarios. Condone \({}^6C_e \times {}^8C_f \times {}^{11}C_g\) with \(e+f+g=7\) to identify S, C, R |
| Total \(= 26964\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(2! \times 3! \times 4! \times 6\) | M1 | \(2! \times 3! \times 4! \times k\), \(k\) an integer \(> 0\). 1 can be implied |
| \(= 1728\) | A1 | If A0 scored SC B1 for 1728 www |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6! \times 7 \times 6 \times 5\) | M1 | \(6! \times k\), \(k\) an integer \(> 0\). 1 can be implied. |
| \(\frac{m!}{a! \times b!} \times 7 \times n \times r\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\); \(1 \leq n\), \(r \leq 6\), \(n \neq r\) |
| \(\frac{m!}{a! \times b!} \times 7 \times 6 \times 5\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\) |
| \(151\,200\) | A1 | Condone \(151\,000\). If A0 scored SC B1 for \(151\,200\) www. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6! \times {}^7P_3\) | M1 | \(6! \times k\), \(k\) an integer \(> 0\). 1 can be implied. |
| \(\frac{m!}{a! \times b!} \times {}^7P_q\) or \(\frac{m!}{a! \times b!} \times {}^7C_q \times q!\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\); \(1 \leq q \leq 6\) |
| \(\frac{m!}{a! \times b!} \times {}^7P_3\) or \(\frac{m!}{a! \times b!} \times {}^7C_3 \times 3!\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\) |
| \(151\,200\) | A1 | Condone \(151\,000\). If A0 scored SC B1 for \(151\,200\) www. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6! \times 35 \times 3!\) | M1 | \(6! \times k\), \(k\) an integer \(> 0\). 1 can be implied. |
| \(\frac{m!}{a! \times b!} \times 35 \times q!\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\); \(1 \leq q \leq 3\) |
| \(\frac{m!}{a! \times b!} \times 35 \times 6\) | M1 | \(6 \leq m \leq 9\); \(a = 1, 2\); \(b = 1, 4\) |
| \(151\,200\) | A1 | Condone \(151\,000\). If A0 scored SC B1 for \(151\,200\) www. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(9! - 7!3! - {}^3P_2 \times 6! \times 7 \times 6\) or \(9! - 7!3! - 3! \times 7! \times 6\) | M1 | \(9! - 7!r! - q\), \(r\) an integer \(> 1\), \(q\) an integer \(\leq 0\). 0 and 1 may be implied. |
| \(\frac{s!}{a! \times b! \times c!} - 7!3! - q\) | M1 | \(s = 8, 9\); \(a = 1, 2\); \(b = 1, 3\); \(c = 1, 4\); \(q\) an integer \(\geq 0\) |
| \(\frac{s!}{a! \times b! \times c!} - 7!3! - {}^3P_2 \times 6! \times 6 \times 7\) or \(\frac{s!}{a! \times b! \times c!} - 7!3! - 3! \times 7! \times 6\) | M1 | \(6 \leq s \leq 9\); \(a = 1, 2\); \(b = 1, 3\); \(c = 1, 4\) |
| \([= 362\,880 - 30\,240 - 181\,440]\) | ||
| \(151\,200\) | A1 | Condone \(151\,000\). If A0 scored SC B1 for \(151\,200\) www. |
| 4 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $S+4C+2R$: ${}^6C_1 \times {}^8C_4 \times {}^{11}C_2 [= 6 \times 70 \times 55] = 23100$ | M1 | ${}^6C_e \times {}^8C_f \times {}^{11}C_g$, with $e+f+g=7$ seen |
| $S+5C+1R$: ${}^6C_1 \times {}^8C_5 \times {}^{11}C_1 [= 6 \times 56 \times 11] = 3696$ | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified, www |
| $S+6C[+0R]$: ${}^6C_1 \times {}^8C_6 \times [{}^{11}C_0][= 6 \times 28] = 168$ | M1 | Add values of 3 correct scenarios. No incorrect scenarios, no repeated scenarios. Condone ${}^6C_e \times {}^8C_f \times {}^{11}C_g$ with $e+f+g=7$ to identify S, C, R |
| Total $= 26964$ | A1 | cao |
**Total: 4 marks**
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $2! \times 3! \times 4! \times 6$ | M1 | $2! \times 3! \times 4! \times k$, $k$ an integer $> 0$. 1 can be implied |
| $= 1728$ | A1 | If A0 scored **SC B1** for 1728 www |
**Total: 2 marks**
## Question 6(c):
**Method 1:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6! \times 7 \times 6 \times 5$ | M1 | $6! \times k$, $k$ an integer $> 0$. 1 can be implied. |
| $\frac{m!}{a! \times b!} \times 7 \times n \times r$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$; $1 \leq n$, $r \leq 6$, $n \neq r$ |
| $\frac{m!}{a! \times b!} \times 7 \times 6 \times 5$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$ |
| $151\,200$ | A1 | Condone $151\,000$. If A0 scored **SC B1** for $151\,200$ www. |
**Method 2:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6! \times {}^7P_3$ | M1 | $6! \times k$, $k$ an integer $> 0$. 1 can be implied. |
| $\frac{m!}{a! \times b!} \times {}^7P_q$ or $\frac{m!}{a! \times b!} \times {}^7C_q \times q!$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$; $1 \leq q \leq 6$ |
| $\frac{m!}{a! \times b!} \times {}^7P_3$ or $\frac{m!}{a! \times b!} \times {}^7C_3 \times 3!$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$ |
| $151\,200$ | A1 | Condone $151\,000$. If A0 scored **SC B1** for $151\,200$ www. |
**Method 3:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6! \times 35 \times 3!$ | M1 | $6! \times k$, $k$ an integer $> 0$. 1 can be implied. |
| $\frac{m!}{a! \times b!} \times 35 \times q!$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$; $1 \leq q \leq 3$ |
| $\frac{m!}{a! \times b!} \times 35 \times 6$ | M1 | $6 \leq m \leq 9$; $a = 1, 2$; $b = 1, 4$ |
| $151\,200$ | A1 | Condone $151\,000$. If A0 scored **SC B1** for $151\,200$ www. |
**Method 4:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $9! - 7!3! - {}^3P_2 \times 6! \times 7 \times 6$ **or** $9! - 7!3! - 3! \times 7! \times 6$ | M1 | $9! - 7!r! - q$, $r$ an integer $> 1$, $q$ an integer $\leq 0$. 0 and 1 may be implied. |
| $\frac{s!}{a! \times b! \times c!} - 7!3! - q$ | M1 | $s = 8, 9$; $a = 1, 2$; $b = 1, 3$; $c = 1, 4$; $q$ an integer $\geq 0$ |
| $\frac{s!}{a! \times b! \times c!} - 7!3! - {}^3P_2 \times 6! \times 6 \times 7$ **or** $\frac{s!}{a! \times b! \times c!} - 7!3! - 3! \times 7! \times 6$ | M1 | $6 \leq s \leq 9$; $a = 1, 2$; $b = 1, 3$; $c = 1, 4$ |
| $[= 362\,880 - 30\,240 - 181\,440]$ | | |
| $151\,200$ | A1 | Condone $151\,000$. If A0 scored **SC B1** for $151\,200$ www. |
| | **4** | |6 In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition.
\begin{enumerate}[label=(\alph*)]
\item Find the number of different ways in which the team of 7 can be selected if it consists of exactly 1 swimmer, at least 4 cyclists and at most 2 runners.\\
For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph.
\item How many different arrangements are there of the 9 people if the swimmers stand together, the cyclists stand together and the runners stand together?
\item How many different arrangements are there of the 9 people if none of the cyclists stand next to each other?\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2023 Q6 [10]}}