| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Digit arrangements forming numbers |
| Difficulty | Moderate -0.8 This is a straightforward permutations question requiring basic counting principles. Part (i) is direct application of 5! = 120. Part (ii)(a) requires recognizing that 3 of 5 digits are odd (simple fraction). Part (ii)(b) needs systematic counting of cases starting with 1 or starting with 2 followed by 1, which is routine but requires careful enumeration. No novel insight needed, just methodical application of basic counting techniques. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| \(120\) | B1 | 1 mark; not just 5! |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 \times 4! = 72\) (\(\div 5!\)) | M1 | |
| \(\frac{1}{5}\) oe | A1 | oe, eg \(\frac{1}{20}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Starts 1 or 21 (both) | M1 | 12, 13, 14, 15, (\(\geq 2\) of these 21), or allow 1 extra; can be implied by wking; 4! + 3! (\(\div 5!\)) complement: full equiv steps for Ms |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{5} + \frac{1}{5} \times \frac{1}{4}\) | M1 | |
| \(= \frac{1}{4}\) oe | A1 | 3 marks |
**i)**
$120$ | B1 | 1 mark; not just 5!
**ii a)**
$3 \times 4! = 72$ ($\div 5!$) | M1 |
$\frac{1}{5}$ oe | A1 | oe, eg $\frac{1}{20}$
**ii b)**
Starts 1 or 21 (both) | M1 | 12, 13, 14, 15, ($\geq 2$ of these 21), or allow 1 extra; can be implied by wking; 4! + 3! ($\div 5!$) complement: full equiv steps for Ms
**ii c)**
$\frac{1}{5} + \frac{1}{5} \times \frac{1}{4}$ | M1 |
$= \frac{1}{4}$ oe | A1 | 3 marks
**Total for Question 3:** 6 marks3 The digits 1, 2, 3, 4 and 5 are arranged in random order, to form a five-digit number.\\
(i) How many different five-digit numbers can be formed?\\
(ii) Find the probability that the five-digit number is
\begin{enumerate}[label=(\alph*)]
\item odd,
\item less than 23000 .
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2007 Q3 [6]}}