| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Seating arrangements with constraints |
| Difficulty | Moderate -0.8 This is a straightforward permutations question with standard techniques: (i) basic arrangement P(17,11), (ii) conditional arrangement with fixed positions, (iii) combination counting with cases. All parts use direct application of formulas without requiring problem-solving insight or complex reasoning—easier than average A-level. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(_{17}P_{11}\) | B1 | Or equivalent |
| \(= 4.94 \times 10^{11}\) | B1 | 2 Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(_{12}P_6 \times 5!\) | B1 | For \(5!\) multiplied by something |
| B1 | For \(_{12}P_6\) or \(_{12}C_6\) multiplied by something | |
| \(= 79800000\ (79833600)\) | B1 | 3 Correct answer o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(_3C_2 \times {}_7C_1\) | B1 | \(3\) or \(_3C_2\) seen |
| M1 | \(_7C\) something seen | |
| \(= 21\) | A1 | 3 Correct answer |
## Question 4(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $_{17}P_{11}$ | B1 | Or equivalent |
| $= 4.94 \times 10^{11}$ | B1 | **2** Or equivalent |
## Question 4(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $_{12}P_6 \times 5!$ | B1 | For $5!$ multiplied by something |
| | B1 | For $_{12}P_6$ or $_{12}C_6$ multiplied by something |
| $= 79800000\ (79833600)$ | B1 | **3** Correct answer o.e. |
## Question 4(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $_3C_2 \times {}_7C_1$ | B1 | $3$ or $_3C_2$ seen |
| | M1 | $_7C$ something seen |
| $= 21$ | A1 | **3** Correct answer |
---4\\
\includegraphics[max width=\textwidth, alt={}, center]{14e8a601-2180-4491-9336-cafd262f2596-3_277_682_274_733}
The diagram shows the seating plan for passengers in a minibus, which has 17 seats arranged in 4 rows. The back row has 5 seats and the other 3 rows have 2 seats on each side. 11 passengers get on the minibus.\\
(i) How many possible seating arrangements are there for the 11 passengers?\\
(ii) How many possible seating arrangements are there if 5 particular people sit in the back row?
Of the 11 passengers, 5 are unmarried and the other 6 consist of 3 married couples.\\
(iii) In how many ways can 5 of the 11 passengers on the bus be chosen if there must be 2 married couples and 1 other person, who may or may not be married?
\hfill \mbox{\textit{CAIE S1 2006 Q4 [8]}}