| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | People arrangements in lines |
| Difficulty | Moderate -0.8 This is a straightforward permutations question testing standard techniques: (i) basic factorial, (ii) gaps method for non-adjacent arrangements, (iii) complementary counting for 'at least one'. All are textbook exercises requiring direct application of formulas with no novel problem-solving or insight needed. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| B1, B1 | 2 marks | \(9!\) Or \(_9P_9\) only. Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | 6! seen | |
| M1 | \(_P\) or \(_C\) something or 7 multiplied by something | |
| A1 | Mult by \(_P_3\) | |
| A1 | 4 marks | Correct answer |
Total \(= 64\)
| Answer | Marks | Guidance |
|---|---|---|
| M1, B1 | Summing cases for 1, 2, 3 women. One correct case | |
| A1 | Correct answer | |
| B1 | \(_3C_3\) or \(84\) or 3 times \(_4C_3\) seen. Attempt at subst of their 'no women' case | |
| M1 | ||
| A1 | 3 marks | Correct answer |
**(i)** $9! = 362880$ $(363000)$
| B1, B1 | 2 marks | $9!$ Or $_9P_9$ only. Correct answer |
**(ii)** $6! \times _5P_3$
$= 151200$
| B1 | 6! seen |
| M1 | $_P$ or $_C$ something or 7 multiplied by something |
| A1 | Mult by $_P_3$ |
| A1 | 4 marks | Correct answer |
**(iii)** 1 woman: $_3C_1 \times {}_4C_2 = 45$
2 women: $_3C_2 \times {}_4C_1 = 18$
3 women: $_3C_3 = 1$
Total $= 64$
OR no restrictions $_9C_3 (84)$ Men only $84 - 20 = 64$
| M1, B1 | Summing cases for 1, 2, 3 women. One correct case |
| A1 | Correct answer |
| B1 | $_3C_3$ or $84$ or 3 times $_4C_3$ seen. Attempt at subst of their 'no women' case |
| M1 | |
| A1 | 3 marks | Correct answer |
---6 Six men and three women are standing in a supermarket queue.\\
(i) How many possible arrangements are there if there are no restrictions on order?\\
(ii) How many possible arrangements are there if no two of the women are standing next to each other?\\
(iii) Three of the people in the queue are chosen to take part in a customer survey. How many different choices are possible if at least one woman must be included?
\hfill \mbox{\textit{CAIE S1 2006 Q6 [9]}}