| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | March |
| 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 multi-part combinations question requiring systematic case-work in part (a), basic permutations with grouping in part (b), and conditional arrangements in part (c). All techniques are routine for S1 level 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 |
| 4M 1W: \({}^6C_4 \times {}^4C_1 = 60\) | M1 | \({}^6C_a \times {}^4C_b\) with \(a + b = 5\) seen, no extra terms. |
| 3M 2W: \({}^6C_3 \times {}^4C_2 = 120\) | B1 | Correct outcome/value for one clearly identified scenario. Accept unsimplified, www. Condone use of \(\times {}^5C_0\). |
| 2M 3W: \({}^6C_2 \times {}^4C_3 = 60\) | M1 | Add values of three correct scenarios, no incorrect scenarios, no repeated scenarios. Condone \({}^6C_a \times {}^4C_b\) with \(a+b=5\) to identify M, W for this mark. |
| Total 240 | A1 | Not dependent on B1. If A0 scored, SC B1 for 240 www. |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(6! \times 4! \times 2\) | M1 | \(6! \times 4! \times k;\ k = 1, 2\). 1 can be implied. |
| \(= 34560\) | A1 | Cao. If M0 scored, SC B1 for 34560 www. |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8! \times 3 \times 2\) | M1 | \(8! \times g,\ g\) an integer greater than 1. |
| M1 | \(h! \times 3 \times j;\ h = 7, 8, 9;\ j = 1, 2\) (1 may be implied). Condone \({}^3C_1\) for 3. | |
| M1 | \(h! \times 3 \times 2;\ h = 7, 8, 9\). Condone \({}^2C_1\) for 2. (Condone \(h! \times 3!\) for M1M1). | |
| \(= 241920\) | A1 | If A0 scored, SC B1 for 241920. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \({}^8C_2 \times 6! \times 3! \times 2\) | M1 | \({}^8C_a \times d,\ a = 2, 6,\ d\) an integer greater than 1. |
| M1 | \(6! \times e,\ e\) an integer greater than 1. | |
| M1 | \({}^8C_a \times f! \times 3! \times 2\) or \({}^8C_a \times f! \times 6 \times 2;\ a = 2, 6;\ f = 5, 6, 7.\) | |
| \(= 241920\) | A1 | If A0 scored, SCB1 for 241920. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \({}^8P_2 \times 6! \times 3!\) | M1 | \({}^8P_2 \times d,\ d\) an integer greater than 1. |
| M1 | \(6! \times e,\ e\) an integer greater than 1. | |
| M1 | \({}^8P_2 \times h! \times 3!\) or \({}^8P_2 \times h! \times 6;\ h = 5, 6, 7.\) | |
| \(= 241920\) | A1 | If A0 scored, SC B1 for 241920. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(^8P_6 \times 3! \times 2!\) | M1 | \(^8P_6 \times d\), \(d\) an integer greater than 1. |
| M1 | \(3! \times e\), \(e\) an integer greater than 1. | |
| M1 | \(^8P_6 \times j! \times 2\); \(j = 1, 2, 3\). | |
| \(= 241920\) | A1 | If A0 Scored, SC B1 for \(241920\). |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 4M 1W: ${}^6C_4 \times {}^4C_1 = 60$ | M1 | ${}^6C_a \times {}^4C_b$ with $a + b = 5$ seen, no extra terms. |
| 3M 2W: ${}^6C_3 \times {}^4C_2 = 120$ | B1 | Correct outcome/value for one clearly identified scenario. Accept unsimplified, www. Condone use of $\times {}^5C_0$. |
| 2M 3W: ${}^6C_2 \times {}^4C_3 = 60$ | M1 | Add values of three correct scenarios, no incorrect scenarios, no repeated scenarios. Condone ${}^6C_a \times {}^4C_b$ with $a+b=5$ to identify M, W for this mark. |
| Total 240 | A1 | Not dependent on B1. If A0 scored, **SC B1** for 240 www. |
| **Total** | **4** | |
---
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $6! \times 4! \times 2$ | M1 | $6! \times 4! \times k;\ k = 1, 2$. 1 can be implied. |
| $= 34560$ | A1 | Cao. If M0 scored, **SC B1** for 34560 www. |
| **Total** | **2** | |
---
## Question 6(c):
**Method 1** – Arrangements of OP in front row, 8 remaining people arranged.
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8! \times 3 \times 2$ | M1 | $8! \times g,\ g$ an integer greater than 1. |
| | M1 | $h! \times 3 \times j;\ h = 7, 8, 9;\ j = 1, 2$ (1 may be implied). Condone ${}^3C_1$ for 3. |
| | M1 | $h! \times 3 \times 2;\ h = 7, 8, 9$. Condone ${}^2C_1$ for 2. (Condone $h! \times 3!$ for M1M1). |
| $= 241920$ | A1 | If A0 scored, **SC B1** for 241920. |
**Method 2** – Two additional people selected for front row, front row arranged, remaining 6 people arranged in back row.
| Answer | Marks | Guidance |
|--------|-------|----------|
| ${}^8C_2 \times 6! \times 3! \times 2$ | M1 | ${}^8C_a \times d,\ a = 2, 6,\ d$ an integer greater than 1. |
| | M1 | $6! \times e,\ e$ an integer greater than 1. |
| | M1 | ${}^8C_a \times f! \times 3! \times 2$ or ${}^8C_a \times f! \times 6 \times 2;\ a = 2, 6;\ f = 5, 6, 7.$ |
| $= 241920$ | A1 | If A0 scored, **SCB1** for 241920. |
**Method 3** – Arrangements of two additional people for front row, front row arranged, remaining 6 people arranged in back row.
| Answer | Marks | Guidance |
|--------|-------|----------|
| ${}^8P_2 \times 6! \times 3!$ | M1 | ${}^8P_2 \times d,\ d$ an integer greater than 1. |
| | M1 | $6! \times e,\ e$ an integer greater than 1. |
| | M1 | ${}^8P_2 \times h! \times 3!$ or ${}^8P_2 \times h! \times 6;\ h = 5, 6, 7.$ |
| $= 241920$ | A1 | If A0 scored, **SC B1** for 241920. |
## Question 6(c):
**Method 4** – Arrangements of 6 people for back row, front row arranged.
| Answer | Mark | Guidance |
|--------|------|----------|
| $^8P_6 \times 3! \times 2!$ | M1 | $^8P_6 \times d$, $d$ an integer greater than 1. |
| | M1 | $3! \times e$, $e$ an integer greater than 1. |
| | M1 | $^8P_6 \times j! \times 2$; $j = 1, 2, 3$. |
| $= 241920$ | A1 | If A0 Scored, **SC B1** for $241920$. |
**Total: 4 marks**6 A new village social club has 10 members of whom 6 are men and 4 are women. The club committee will consist of 5 members.
\begin{enumerate}[label=(\alph*)]
\item In how many ways can the committee of 5 members be chosen if it must include at least 2 men and at least 1 woman?\\
The 10 members of the club stand in a line for a photograph.
\item How many different arrangements are there of the 10 members if all the men stand together and all the women stand together?\\
For a second photograph, the members stand in two rows, with 6 on the back row and 4 on the front row. Olly and his sister Petra are two of the members of the club.
\item How many different arrangements are there of the 10 members in which Olly and Petra stand next to each other on the front row?\\
If you use the following 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 2024 Q6 [10]}}