| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | People arrangements in groups/rows |
| Difficulty | Standard +0.3 Part (a) requires systematic case enumeration (pairs summing to 4) and recognizing the middle 5 dice are unrestricted (6^5), which is straightforward counting. Part (b) involves partitioning 9 items into three odd groups (1-1-7, 1-3-5, 3-3-3) with multinomial coefficients—slightly above routine but still standard S1 combinatorics with clear structure. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \( | * * * * * 3\) or \(3 * * * * * | \) or \(2 * * * * * ?\) |
| \(= 6^5 \times 3\) | M1 | Mult by 3 or summing 3 different combinations (for end dice outcomes) |
| \(= 23328\) | A1 3 | Correct answer accept 23 300 |
| (b) W | J | H |
| 1 | 1 | \(7 = ^3C_1 \times ^5C_1 \times 1 = 72\) |
| 1 | 7 | \(1 = ^5C_1 \times ^3C_1 \times 1 = 72\) |
| 7 | 1 | \(1 = ^7C_1 \times ^3C_1 \times 1 = 72\) |
| 1 | 3 | \(5 = ^3C_1 \times ^5C_3 \times 1 = 504\) mult by 3! |
| 3 | 3 | \(3 = ^3C_3 \times ^5C_3 \times 1 = 1680\) |
| M1 | Summing at least 2 different options of the 3 | |
| A1 6 | Correct ans | |
| If no marks gained | SCM1 | If games replaced M1M1M1 max available |
| Listing all 10 different outcomes | If factorials used M0M1M1 max available |
**(a)** $|* * * * * 3$ or $3 * * * * * |$ or $2 * * * * * ?$ | M1 | Mult by 6$^5$ (for middle 5 dice outcomes)
$= 6^5 \times 3$ | M1 | Mult by 3 or summing 3 different combinations (for end dice outcomes)
$= 23328$ | A1 3 | Correct answer accept 23 300
**(b)** W | J | H |
1 | 1 | $7 = ^3C_1 \times ^5C_1 \times 1 = 72$ | M1 | Multiplying 3 combinations (may be implied)
1 | 7 | $1 = ^5C_1 \times ^3C_1 \times 1 = 72$ | A1 | 1 unsimplified correct answer (72, 504, 1680, 216 or 3024)
7 | 1 | $1 = ^7C_1 \times ^3C_1 \times 1 = 72$ | A1 |
1 | 3 | $5 = ^3C_1 \times ^5C_3 \times 1 = 504$ mult by 3! | A1 A 2nd unsimplified different correct answer
3 | 3 | $3 = ^3C_3 \times ^5C_3 \times 1 = 1680$ | M1 | Summing options for 1,1,7 or 1,3,5 oe (mult by 3 or 3!)
| M1 | Summing at least 2 different options of the 3
| A1 6 | Correct ans
If no marks gained | SCM1 | If games replaced M1M1M1 max available
Listing all 10 different outcomes | | If factorials used M0M1M1 max available6
\begin{enumerate}[label=(\alph*)]
\item Seven fair dice each with faces marked 1,2,3,4,5,6 are thrown and placed in a line. Find the number of possible arrangements where the sum of the numbers at each end of the line add up to 4 .
\item Find the number of ways in which 9 different computer games can be shared out between Wainah, Jingyi and Hebe so that each person receives an odd number of computer games.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2014 Q6 [9]}}