| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Standard +0.3 This is a standard combinations problem with constraints requiring systematic case-by-case enumeration (part i) and a straightforward application of the division principle (part ii). While it requires careful organization and multiple calculations, the techniques are routine for A-level statistics students and involve no novel problem-solving insights. |
| Spec | 5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| \(3A\ 3D\ 1M: {}^6C_3 \times {}^5C_3 \times {}^4C_1 (= 800)\) | M1 | \({}^6C_x \times {}^5C_y \times {}^4C_z,\ x+y+z=7\) |
| A1 | 2 correct products, allow unsimplified | |
| M1 | Summing their totals for 3 correct scenarios only | |
| \(\text{Total} = 2600\) | A1 | Correct answer; SC1 \({}^6C_3 \times {}^5C_2 \times {}^4C_1 \times {}^9C_1 = 7200\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(^7C_4 \times 1\) | B1 | \(^7C_3\) or \(^7C_4\) seen anywhere |
| \(35\) | B1 | |
| 2 |
## Question 3:
### Part (i)
$3A\ 2D\ 2M: {}^6C_3 \times {}^5C_2 \times {}^4C_2 (= 1200)$
$4A\ 2D\ 1M: {}^6C_4 \times {}^5C_2 \times {}^4C_1 (= 600)$
$3A\ 3D\ 1M: {}^6C_3 \times {}^5C_3 \times {}^4C_1 (= 800)$ | **M1** | ${}^6C_x \times {}^5C_y \times {}^4C_z,\ x+y+z=7$
| **A1** | 2 correct products, allow unsimplified
| **M1** | Summing their totals for 3 correct scenarios only
$\text{Total} = 2600$ | **A1** | Correct answer; **SC1** ${}^6C_3 \times {}^5C_2 \times {}^4C_1 \times {}^9C_1 = 7200$
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $^7C_4 \times 1$ | B1 | $^7C_3$ or $^7C_4$ seen anywhere |
| $35$ | B1 | |
| | **2** | |
---3 A sports team of 7 people is to be chosen from 6 attackers, 5 defenders and 4 midfielders. The team must include at least 3 attackers, at least 2 defenders and at least 1 midfielder.\\
\begin{enumerate}[label=(\roman*)]
\item In how many different ways can the team of 7 people be chosen?\\
The team of 7 that is chosen travels to a match in two cars. A group of 4 travel in one car and a group of 3 travel in the other car.
\item In how many different ways can the team of 7 be divided into a group of 4 and a group of 3 ?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2019 Q3 [6]}}