| Exam Board | OCR |
|---|---|
| Module | FD1 AS (Further Decision 1 AS) |
| Year | 2018 |
| Session | March |
| Marks | 10 |
| Topic | Permutations & Arrangements |
| Type | Basic arrangements with repeated letters |
| Difficulty | Standard +0.3 This is a multi-part question testing standard techniques: (i) uses basic factorial divisibility (25! contains factors of 5), (ii) applies the standard algorithm for finding highest power of a prime in n!, and (iii) is a straightforward application of inclusion-exclusion with simple arithmetic. All parts are textbook exercises requiring recall of known methods rather than problem-solving or insight, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities7.01d Multiplicative principle: arrangements of n distinct objects7.01k Inclusion-exclusion: for two sets7.01l Inclusion-exclusion: extended to more than two sets8.01h Modelling with recurrence: birth/death rates, INT function |
1
\begin{enumerate}[label=(\roman*)]
\item (a) Show that the number of arrangements of 25 distinct objects is an integer multiple of $5 ^ { 6 }$.\\
(b) Explain how this shows that the number of arrangements of 25 distinct objects is a whole number of millions.
\item (a) Calculate the values of
\begin{itemize}
\item INT(720 $\div 25$ )
\item INT(720 $\div 125$ ).\\
(b) Deduce the largest power of 10 that is a factor of 720!
\item Use the inclusion-exclusion principle to find the number of integers from 1 to 720 that are not divisible by either 2 or 5 .
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR FD1 AS 2018 Q1 [10]}}