| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2004 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Arrangements with alternating patterns |
| Difficulty | Moderate -0.3 Part (i) is a straightforward application of the formula for permutations with repeated letters (9!/3! for the three A's). Part (ii) requires recognizing that consonants and vowels must alternate, then counting arrangements of each group separately (4! × 3!), which is a standard technique but requires one additional conceptual step beyond basic permutations. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{9!}{2!2!} = 90720\) | B1 | For dividing by \(2!\) or \(2\) once or twice, or \({}_{9}P_7\) or \({}_{9}C_7\) seen, can be implied |
| \(90720\) | B1 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{5!4!}{2!2!} = 720\) OR could do by probs and multiply by (i) | B1 | For \(5!\) or \(4!\) or \({}_{4}P_4\) or \({}_{5}P_5\) seen in numerator |
| B1 | For \(5! \times 4! \times k\) in numerator of a term, \(k = 1\) or \(2\) only | |
| \(720\) | B1 | 3 |
## Question 1:
### Part (i)
$\frac{9!}{2!2!} = 90720$ | B1 | For dividing by $2!$ or $2$ once or twice, or ${}_{9}P_7$ or ${}_{9}C_7$ seen, can be implied
$90720$ | B1 | **2** | For correct answer
### Part (ii)
$\frac{5!4!}{2!2!} = 720$ OR could do by probs and multiply by (i) | B1 | For $5!$ or $4!$ or ${}_{4}P_4$ or ${}_{5}P_5$ seen in numerator
| B1 | For $5! \times 4! \times k$ in numerator of a term, $k = 1$ or $2$ only
$720$ | B1 | **3** | For correct final answer
---1 The word ARGENTINA includes the four consonants R, G, N, T and the three vowels A, E, I.\\
(i) Find the number of different arrangements using all nine letters.\\
(ii) How many of these arrangements have a consonant at the beginning, then a vowel, then another consonant, and so on alternately?
\hfill \mbox{\textit{CAIE S1 2004 Q1 [5]}}