| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Graph theory problems |
| Difficulty | Standard +0.3 This is a straightforward application of counting arrangements with precedence constraints. Part (i) requires systematic enumeration of 4 activities with simple ordering rules, part (ii) guides students through the reasoning for 5 activities, and part (iii) extends to 6 activities. The constraints are clearly stated and the problem requires careful case-by-case counting rather than novel insight. This is slightly easier than average as it's a guided, multi-part question with explicit structure and the 'explain carefully' prompts scaffold the solution method. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (i) | 3 possible orders: ABCD ACBD CABD |
| [1] | 1.1 | n |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (ii) | E must come after C, but otherwise it can go |
| Answer | Marks |
|---|---|
| In total there are 9 possible orders | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.4 | e |
| Answer | Marks |
|---|---|
| more | A B C E D A B C D E |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (iii) | F must come after C |
| Answer | Marks |
|---|---|
| In total there are 38 possible orders | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 2.4 | Seen, or implied by being true for all |
| Answer | Marks |
|---|---|
| With appropriate working | A listing of the possibilities, |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | 1 | 1 |
| 2 | 0 | 2 |
| 2 | 7 | 8 |
Question 2:
2 | (i) | 3 possible orders: ABCD ACBD CABD | B1
[1] | 1.1 | n
Stating 3 or listing the three orders
2 | (ii) | E must come after C, but otherwise it can go
anywhere
If A, B, C, D are in the order A B C D then E
can go immediately after C or D (cid:159)2
If they are in the order A C B D then E can go
immediately after C, B or D (cid:159)3
If they are in the order C A B D then E can go
immediately after C, A, B or D (cid:159)4
p
In total there are 9 possible orders | B1
M1
e
E1
[3] | 1.1
2.1
i
c
2.4 | e
m
Seen, or implied by being true for all
given orders
Breaking the problem up into cases
Showing that there are 9 orders and no
more | A B C E D A B C D E
A C E B D A C B E D
A C B D E C E A B D
C A E B D C A B E D
C A B D E
2 | (iii) | F must come after C
If C is in the third position (A B C E D,
A B C D E) there are 3 possibilities for F (cid:159)6
If C is in the second position (A C E B D,
A C B E D, A C B D E) there are 4 possibilities
for F (cid:159)12
If C is in the first position (C E A B D,
C A E B D, C A B E D, C A B D E) there are 5
possibilities for F (cid:159)20
In total there are 38 possible orders | B1
M1
E1
[3] | 1.1
2.1
2.4 | Seen, or implied by being true for all
given orders
Making a substantial start at counting
the possibilities, this may involve
starting again
n
e
With appropriate working | A listing of the possibilities,
without written reasoning,
would score M0
2 | 1 | 1 | 7.50
2 | 0 | 2 | 9.00
2 | 7 | 8 | 5 | -
--- 2(i) ---
2(i)
n
e
--- 2(ii) ---
2(ii)
m
i
c
e
p
S
--- 2(iii) ---
2(iii)2 Some of the activities that may be involved in making a cup of tea are listed below.
A: Boil water.\\
B: Put teabag in teapot, pour on boiled water and let tea brew.\\
C: Get cup from cupboard.\\
D: Pour tea into cup.\\
E: Add milk to cup.\\
F: Add sugar to cup.
Activity A must happen before activity B.\\
Activities B and C must happen before activity D .\\
Activities E and F cannot happen until after activity C.\\
Other than that, the activities can happen in any order.\\
(i) Lisa does not take milk or sugar in her tea, so she only needs to use activities $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D . In how many different orders can activities $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D be arranged, subject to the restrictions above?\\
(ii) Mick takes milk but no sugar, so he needs to use activities $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }$ and E . Explain carefully why there are exactly nine different orders for these activities, subject to the restrictions above.\\
(iii) Find the number of different orders for all six activities, subject to the restrictions above. Explain your reasoning carefully.
\hfill \mbox{\textit{OCR Further Discrete AS Q2 [7]}}