| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Arrangements in multiple rows/groups |
| Difficulty | Standard +0.3 Part (i) requires understanding that Jon and Sarah must be together, then selecting 2 more from 6 remaining friends for their taxi, with a division by 2 for indistinguishable taxis. Part (ii) involves arranging people with constraints (Mark fixed in front, Jon and Sarah together in back) across two taxis. Both parts use standard combinatorial techniques with mild constraints, slightly easier than average A-level due to straightforward application of selection and arrangement principles. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Two in same taxi: \(^6C_2 \times {^4C_4} \times 2\) or \(^6C_2 + {^6C_4}\) | M1 | \(^6C_4\) or \(^6C_2\) oe seen anywhere |
| M1 | 'something' \(\times 2\) only or adding 2 equal terms | |
| \(= 30\) | A1 (3) | Correct final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| MJS in taxi: \((^5C_1 \times 2 \times 2) \times {^4P_4}\) | M1 | \(^5P_1\), \(^5C_1\) or 5 seen anywhere |
| M1 | Mult by 2 or 4 oe | |
| M1 | Mult by \(^4P_4\) oe e.g. 4! or \(4\times^3P_3\) or can be part of 5! | |
| \(= 480\) | A1 (4) | Correct final answer |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Two in same taxi: $^6C_2 \times {^4C_4} \times 2$ or $^6C_2 + {^6C_4}$ | M1 | $^6C_4$ or $^6C_2$ oe seen anywhere |
| | M1 | 'something' $\times 2$ only or adding 2 equal terms |
| $= 30$ | A1 (3) | Correct final answer |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| MJS in taxi: $(^5C_1 \times 2 \times 2) \times {^4P_4}$ | M1 | $^5P_1$, $^5C_1$ or 5 seen anywhere |
| | M1 | Mult by 2 or 4 oe |
| | M1 | Mult by $^4P_4$ oe e.g. 4! or $4\times^3P_3$ or can be part of 5! |
| $= 480$ | A1 (4) | Correct final answer |
---4 A group of 8 friends travels to the airport in two taxis, $P$ and $Q$. Each taxi can take 4 passengers.\\
(i) The 8 friends divide themselves into two groups of 4, one group for taxi $P$ and one group for taxi $Q$, with Jon and Sarah travelling in the same taxi. Find the number of different ways in which this can be done.\\
\includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_284_467_1491_495}\\
\includegraphics[max width=\textwidth, alt={}, center]{e2f57f0f-d9dd-4506-afdd-77d61bd47e4b-2_286_471_1489_1183}
Each taxi can take 1 passenger in the front and 3 passengers in the back (see diagram). Mark sits in the front of taxi $P$ and Jon and Sarah sit in the back of taxi $P$ next to each other.\\
(ii) Find the number of different seating arrangements that are now possible for the 8 friends.
\hfill \mbox{\textit{CAIE S1 2015 Q4 [7]}}