| Exam Board | OCR |
|---|---|
| Module | Further Discrete (Further Discrete) |
| Year | 2017 |
| Session | Specimen |
| Marks | 9 |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Standard +0.8 This is a Further Maths discrete mathematics question on derangements requiring understanding of combinatorics beyond standard A-level. While individual parts are structured with guidance, it requires proof/explanation of the derangement recurrence relation and multi-step reasoning about constrained permutations. The conceptual demand of derangements and the recurrence relation explanation elevate this above typical A-level questions, though the scaffolding prevents it from being extremely difficult. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.
\begin{enumerate}[label=(\roman*)]
\item How many different ways are there to pair the letters with the envelopes? [1]
\item Find the number of arrangements with exactly three letters in the correct envelopes. [2]
\item \begin{enumerate}[label=(\alph*)]
\item Show that there are two derangements of the three symbols A, B and C. [1]
\item Hence find the number of arrangements with exactly two letters in the correct envelopes. [1]
\end{enumerate}
\end{enumerate}
Let $D_n$ represent the number of derangements of $n$ symbols.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Explain why $D_n = (n-1) \times (D_{n-1} + D_{n-2})$. [2]
\item Find the number of ways in which all five letters are in the wrong envelopes. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete 2017 Q3 [9]}}