| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Committee with gender/category constraints |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the multiplication principle with three independent selections using combinations. Part (b) requires systematic case analysis (considering 0 or 1 pianist, then distributing remaining spots between violinists and guitarists while respecting constraints), which adds modest complexity but remains a standard S1 exercise with clear structure and no novel insight required. |
| Spec | 5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(^{10}C_4 \times {}^6C_2 \times {}^5C_1\) | M1 | \(^{10}C_a \times {}^6C_b \times {}^5C_c\), \(a+b+c=7\), \(a,b,c\) integers. No other terms present but condone \(\times 6\) or \(\times 3!\) |
| \([= 210 \times 15 \times 5] = 15750\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Scenarios: VVVVG: 4 1 0 → \(^{10}C_4 \times {}^6C_1\ [\times {}^5C_0]\) \([1260]\); VVVGG: 3 2 0 → \(^{10}C_3 \times {}^6C_2\ [\times {}^5C_0]\) \([1800]\); VVGGG: 2 3 0 → \(^{10}C_2 \times {}^6C_3\ [\times {}^5C_0]\) \([900]\); VVVGP: 3 1 1 → \(^{10}C_3 \times {}^6C_1 \times {}^5C_1\) \([3600]\); VVGGP: 2 2 1 → \(^{10}C_2 \times {}^6C_2 \times {}^5C_1\) \([3375]\) | M1 | One product using 2 or 3 combinations with upper numbers correct and lower numbers summing to 5 and linked to a correct identified scenario. Condone consistent use of permutations |
| 2 identified outcomes evaluated accurately | B1 | Accept unsimplified |
| Add values of 5 correct scenarios, no incorrect/repeated scenarios | M1 | |
| Total \(= 10935\) | A1 | If either or both Ms not awarded, SC B1 for 10935 WWW |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $^{10}C_4 \times {}^6C_2 \times {}^5C_1$ | M1 | $^{10}C_a \times {}^6C_b \times {}^5C_c$, $a+b+c=7$, $a,b,c$ integers. No other terms present but condone $\times 6$ or $\times 3!$ |
| $[= 210 \times 15 \times 5] = 15750$ | A1 | |
**Total: 2 marks**
---
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Scenarios: VVVVG: 4 1 0 → $^{10}C_4 \times {}^6C_1\ [\times {}^5C_0]$ $[1260]$; VVVGG: 3 2 0 → $^{10}C_3 \times {}^6C_2\ [\times {}^5C_0]$ $[1800]$; VVGGG: 2 3 0 → $^{10}C_2 \times {}^6C_3\ [\times {}^5C_0]$ $[900]$; VVVGP: 3 1 1 → $^{10}C_3 \times {}^6C_1 \times {}^5C_1$ $[3600]$; VVGGP: 2 2 1 → $^{10}C_2 \times {}^6C_2 \times {}^5C_1$ $[3375]$ | M1 | One product using 2 or 3 combinations with upper numbers correct and lower numbers summing to 5 and linked to a correct identified scenario. Condone consistent use of permutations |
| 2 identified outcomes evaluated accurately | B1 | Accept unsimplified |
| Add values of 5 correct scenarios, no incorrect/repeated scenarios | M1 | |
| Total $= 10935$ | A1 | If either or both Ms not awarded, **SC B1** for 10935 WWW |
**Total: 4 marks**
---5 In a class of 21 students, there are 10 violinists, 6 guitarists and 5 pianists. A group of 7 is to be chosen from these 21 students. The group will consist of 4 violinists, 2 guitarists and 1 pianist.
\begin{enumerate}[label=(\alph*)]
\item In how many ways can the group of 7 be chosen?\\
On another occasion a group of 5 will be chosen from the 21 students. The group must contain at least 2 violinists, at least 1 guitarist and at most 1 pianist.
\item In how many ways can the group of 5 be chosen?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q5 [6]}}