| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Items NOT together (general separation) |
| Difficulty | Moderate -0.8 This is a straightforward permutations question with standard techniques: (i) is simple multiplication principle (4×3×7), (ii) is basic 'not together' arrangement using complement (10! - 9!×2!), and (iii) is simple combinations summation (2^9 - 1). All parts require only direct application of standard formulas with no problem-solving insight, making it easier than average but not trivial due to the multi-part nature. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(4 \times 3 \times 7 = 84\) | B1 [1] | Correct answer |
| (ii) \(10! - 9! \times 2 = 2903040 (2900000)\) | B1, B1 [2] | \(10! - k \times 9!\) seen or; Correct answer |
| OR \(8! \times 9 \times 8 = 2903040 (2900000)\) | B1, B1 | \(8! \times 9 \times 8\) / seen or; Correct answer |
| (iii) \(^9C_1 + ^9C_2 + ... + ^9C_9\) | M1, M1 | Using combinations; Adding 9 combinations |
| \(= 511\) | A1 [3] | Correct answer |
| OR \(2^9 - 1\) | M1, M1 | \(2^9\) seen; Subtracting 1 |
| \(= 511\) | A1 | Correct answer |
**(i)** $4 \times 3 \times 7 = 84$ | B1 [1] | Correct answer
**(ii)** $10! - 9! \times 2 = 2903040 (2900000)$ | B1, B1 [2] | $10! - k \times 9!$ seen or; Correct answer
OR $8! \times 9 \times 8 = 2903040 (2900000)$ | B1, B1 | $8! \times 9 \times 8$ / seen or; Correct answer
**(iii)** $^9C_1 + ^9C_2 + ... + ^9C_9$ | M1, M1 | Using combinations; Adding 9 combinations
$= 511$ | A1 [3] | Correct answer
OR $2^9 - 1$ | M1, M1 | $2^9$ seen; Subtracting 1
$= 511$ | A1 | Correct answer
---2 Fahad has 4 different coloured pairs of shoes (white, red, blue and black), 3 different coloured pairs of jeans (blue, black and brown) and 7 different coloured tee shirts (red, orange, yellow, blue, green, white and purple).\\
(i) Fahad chooses an outfit consisting of one pair of shoes, one pair of jeans and one tee shirt. How many different outfits can he choose?\\
(ii) How many different ways can Fahad arrange his 3 jeans and 7 tee shirts in a row if the two blue items are not next to each other?
Fahad also has 9 different books about sport. When he goes on holiday he chooses at least one of these books to take with him.\\
(iii) How many different selections are there if he can take any number of books ranging from just one of them to all of them?
\hfill \mbox{\textit{CAIE S1 2011 Q2 [6]}}