| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Digit arrangements forming numbers |
| Difficulty | Easy -1.2 This is a straightforward counting problem requiring systematic enumeration of cases (PINs starting 0, all same digits, consecutive sequences) and subtraction from 10,000 total PINs, followed by a simple division. The arithmetic is routine and no sophisticated combinatorial insight is needed—just careful case-by-case counting, making it easier than average A-level questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | correct binomial coefficients | |
| A1 | \(1 + x - ..\) | |
| A1 | ||
| M1 | or \(\ | 3x\ |
| A1 | ||
| ie 1, 1/3, (1/3)(-2/3)/2 not nCr form simplified www in this part simplified www in this part, ignore subsequent terms using (3x)³ as 3x² can score M1B1B0 condone omission of brackets if 3x² is used as 9x² do not allow MR for power 3 or -1/3 or similar condone inequality signs throughout or say < at one end and ≤ at the other condone -1/3 ≤ \ | x\ | ≤ 1/3, x<1/3 is M0A0 the last two marks are not dependent on the first three |
**Answer:** $(1+3x)^{-1} = 1 + \frac{1}{3}(3x) + \frac{1}{2!}\cdot\frac{2}{3^2}(3x)^2 + ...$
$= 1 + x - x^2 + ...$
Valid for $-1 \leq 3x \leq 1$
$\Rightarrow -\frac{1}{3} \leq x \leq \frac{1}{3}$
| M1 | correct binomial coefficients |
| A1 | $1 + x - ..$ |
| A1 | |
| M1 | or $\|3x\| \leq 1$ oe or $\|x\| \leq 1/3$ (correct final answer scores M1A1) |
| A1 | |
| | ie 1, 1/3, (1/3)(-2/3)/2 not nCr form simplified www in this part simplified www in this part, ignore subsequent terms using (3x)³ as 3x² can score M1B1B0 condone omission of brackets if 3x² is used as 9x² do not allow MR for power 3 or -1/3 or similar **condone inequality signs throughout or say < at one end and ≤ at the other** **condone -1/3 ≤ \|x\| ≤ 1/3, x<1/3 is M0A0** the last two marks are not dependent on the first three |
**Total: [5]**
---\begin{enumerate}[label=(\roman*)]
\item In lines 8 to 10, the article says "Some banks do not allow numbers that begin with zero, numbers in which the digits are all the same (such as 5555) or numbers in which the digits are consecutive (such as 2345 or 8765)."
How many different 4-digit PINs can be made when all these rules are applied? [3]
\item At the time of writing, the world population is $6.7 \times 10^9$ people. Assuming that, on average, each person has one card with a 4-digit PIN (subject to the rules in part (i) of this question), estimate the average number of people holding cards with any given PIN. Give your answer to an appropriate degree of accuracy. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2011 Q2 [5]}}