| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Challenging +1.2 Part (a) requires systematic case analysis with constraints (at least 11 geraniums, equal roses and lilies summing to 25) and applying combinations correctly across multiple cases. Part (b) involves a non-standard permutation problem requiring careful treatment of repeated letters with a proximity constraint, demanding strategic use of complementary counting or grouping techniques. Both parts require problem-solving beyond routine application of formulas. |
| Spec | 5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| G | R | L |
| 11 | 7 | 7 |
| 13 | 6 | 6 |
| 15 | 5 | 5 |
| Total = 1941912 (1940000) | M1, A1, M1, A1 | Multiplying 3 combinations; One of 1310400, 617400, 14112 seen; Adding 3 options; [4] Correct answer |
| (b) e.g. * E * R * E (GG) N * A * E * gives 6 ways for G: \(\frac{7!}{3!} \times 6\) or 8!/3! − 2 × 7!/3! = 5040 ways | B1, B1, B1 | [3] 7! / 3! Or 7!/3!3! seen or 7!/3!3! seen; Multiplying by 6 (gaps) or; Correct final answer |
**(a)**
| G | R | L |
|---|---|---|
| 11 | 7 | 7 | 15C11 × 10C7 × 8C7 = 1310400
| 13 | 6 | 6 | 15C13 × 10C6 × 8C6 = 617400
| 15 | 5 | 5 | 15C15 × 10C5 × 8C5 = 14112
Total = 1941912 (1940000) | M1, A1, M1, A1 | Multiplying 3 combinations; One of 1310400, 617400, 14112 seen; Adding 3 options; [4] Correct answer
**(b)** e.g. * E * R * E (GG) N * A * E * gives 6 ways for G: $\frac{7!}{3!} \times 6$ or 8!/3! − 2 × 7!/3! = 5040 ways | B1, B1, B1 | [3] 7! / 3! Or 7!/3!3! seen or 7!/3!3! seen; Multiplying by 6 (gaps) or; Correct final answer3
\begin{enumerate}[label=(\alph*)]
\item Geoff wishes to plant 25 flowers in a flower-bed. He can choose from 15 different geraniums, 10 different roses and 8 different lilies. He wants to have at least 11 geraniums and also to have the same number of roses and lilies. Find the number of different selections of flowers he can make.
\item Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged if exactly two of the Gs are next to each other.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2011 Q3 [7]}}