| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2016 |
| Session | Specimen |
| Marks | 11 |
| Topic | Combinations & Selection |
| Type | Committee with gender/category constraints |
| Difficulty | Moderate -0.8 This question combines basic set theory/Venn diagrams with straightforward combinations. Part (a) requires simple probability calculations from a two-way table, while part (b) involves standard combination formulas with minimal constraints. All techniques are routine A-level material with no novel problem-solving required, making it easier than average. |
| Spec | 2.03c Conditional probability: using diagrams/tables5.01a Permutations and combinations: evaluate probabilities |
**Question 2(a)**
(i) $75 - x + x + 130 - x = 170$
$x = 35$ (Finding the intersection) [M1]
State $75 - 35$ o.e. [A1]
$\frac{40}{200}$ o.e. [A1]
(ii) Use conditional probability [M1]
$\frac{\text{their } 35}{\text{their } 130}$
$= \frac{35}{130}$ o.e. [A1]
**(b)(i)** Recognise combination problem [M1]
$^{15}C_7 = \frac{15!}{8!7!} = 6435$ [A1]
(ii) $^6C_2 \times {^9C_5}$ correct method [M1]
$= 1890$ [A1]
(iii) $(6M\ 1C) + (5M\ 2C) + (4M\ 3C)$ correct method [M1]
$^6C_6 \times {^9C_1} + {^6C_5} \times {^9C_2} + {^6C_4} \times {^9C_3}$ [M1]
$= 1485$ [A1]
**Total: 11 marks**2
\begin{enumerate}[label=(\alph*)]
\item A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
\begin{enumerate}[label=(\roman*)]
\item Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
\item Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
\end{enumerate}\item The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
\begin{enumerate}[label=(\roman*)]
\item there are no restrictions,
\item the programme must consist of 5 classical pieces and 2 modern pieces,
\item there are to be more modern pieces than classical pieces.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2016 Q2 [11]}}