| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Selection with family/relationship restrictions |
| Difficulty | Moderate -0.3 This is a straightforward combinations problem with basic restrictions. Part (a) requires simple case-splitting (Mr Lan only OR Mrs Lan only) and part (b) requires counting favorable gender compositions. Both parts use standard C(n,r) calculations with no conceptual difficulty beyond organizing cases—slightly easier than average due to clear structure and routine technique. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(^{12}C_4 \times 2\) | M1 | \(^gC_4 \times h\), \(g = 12, 13\), \(h = 1, 2\) |
| \(990\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([total - both\ on - neither\ on]\ ^{14}C_5 - (^{12}C_3 + ^{12}C_5) = [2002 - 220 - 792]\) | M1 | \(^kC_5 - (^aC_3 + ^aC_5)\), \(a = 12, 13\) and \(k = 13, 14\) |
| \(990\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| [Mrs Lan plus] 2W 2M: \(^7C_2 \times ^6C_2 = 315\); 3W 1M: \(^7C_3 \times ^6C_1 = 210\); 4W: \(^7C_4 = 35\) | M1 | \(^7C_r \times ^6C_{4-r}\) for \(r = 2, 3\) or \(4\) |
| One scenario correct, accept unevaluated | B1 | Outcome for one identifiable scenario correct |
| Add outcomes for 3 identifiable correct scenarios | M1 | Note: if scenarios not labelled, identified by seeing \(^7C_r \times ^6C_s\), \(r + s = 4\) to imply \(r\) women and \(s\) men for both B & M marks only |
| \([Total =]\ 560\) | A1 |
## Question 2(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $^{12}C_4 \times 2$ | M1 | $^gC_4 \times h$, $g = 12, 13$, $h = 1, 2$ |
| $990$ | A1 | |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[total - both\ on - neither\ on]\ ^{14}C_5 - (^{12}C_3 + ^{12}C_5) = [2002 - 220 - 792]$ | M1 | $^kC_5 - (^aC_3 + ^aC_5)$, $a = 12, 13$ and $k = 13, 14$ |
| $990$ | A1 | |
---
## Question 2(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| [Mrs Lan plus] 2W 2M: $^7C_2 \times ^6C_2 = 315$; 3W 1M: $^7C_3 \times ^6C_1 = 210$; 4W: $^7C_4 = 35$ | M1 | $^7C_r \times ^6C_{4-r}$ for $r = 2, 3$ or $4$ |
| One scenario correct, accept unevaluated | B1 | Outcome for one identifiable scenario correct |
| Add outcomes for 3 identifiable correct scenarios | M1 | Note: if scenarios not labelled, identified by seeing $^7C_r \times ^6C_s$, $r + s = 4$ to imply $r$ women and $s$ men for both B & M marks only |
| $[Total =]\ 560$ | A1 | |
---2 There are 6 men and 8 women in a Book Club. The committee of the club consists of five of its members. Mr Lan and Mrs Lan are members of the club.
\begin{enumerate}[label=(\alph*)]
\item In how many different ways can the committee be selected if exactly one of Mr Lan and Mrs Lan must be on the committee?
\item In how many different ways can the committee be selected if Mrs Lan must be on the committee and there must be more women than men on the committee?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q2 [6]}}