| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Pigeonhole principle applications |
| Difficulty | Standard +0.8 This is a Further Maths discrete mathematics question requiring understanding of the pigeonhole principle and its non-trivial application. Part (ii) requires careful proof construction with consecutive day constraints, and part (iii) involves stars-and-bars combinatorics. While the individual techniques are standard for Further Maths, the question demands mathematical maturity and precise reasoning beyond typical A-level problems. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities7.01c Pigeonhole principle7.01g Arrangements in a line: with repetition and restriction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| There must be at least one day on which Mo eats at least two doughnuts | B1 [1] | Or equivalent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| The maximum on any one day is 3; any day next to the 3 must be a 1, \(3+1=4\) | E1 | \(1\ 1\ \underline{1\ 3},\ 1\ \underline{1\ 3}\ 1,\ \underline{1\ 3}\ 1\ 1,\ \underline{3}\ 1\ 1\ 1\) |
| Or 2 on two days and 1 on two days | M1 | Start to consider cases for 2 2 1 1; May be implied from working |
| If the 2's are on adjacent days this gives 4 | ||
| If not there is a 2 on (at least one) end so the other three days total 4 | A1 [4] | May list and identify where total \(= 4\); Dealing with both 2 2 1 1 situations and describing or showing 4's; \(\underline{2\ 2}\ 1\ 1,\ 1\ \underline{2\ 2}\ 1\) or \(1\ 1\ \underline{2\ 2}\); \(\underline{2}\ 1\ 1\ 2,\ 1\ \underline{2}\ 1\ 2\) or \(2\ 1\ \underline{2}\ 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4!\div 2! = 12\) arrangements of 0, 0, 1, 2 | B1 | Or \(^4C_2 \times 2 = 12\) or write out these 12 |
| 4 arrangements of 0, 1, 1, 1 | B1 | Or write out these 4 |
| 4 arrangements of 0, 0, 0, 3 | B1 [3] | Or write out these 4 |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| There must be at least one day on which Mo eats at least two doughnuts | B1 [1] | Or equivalent |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| The maximum on any one day is 3; any day next to the 3 must be a 1, $3+1=4$ | E1 | $1\ 1\ \underline{1\ 3},\ 1\ \underline{1\ 3}\ 1,\ \underline{1\ 3}\ 1\ 1,\ \underline{3}\ 1\ 1\ 1$ |
| Or 2 on two days and 1 on two days | M1 | Start to consider cases for 2 2 1 1; May be implied from working |
| If the 2's are on adjacent days this gives 4 | | |
| If not there is a 2 on (at least one) end so the other three days total 4 | A1 [4] | May list and identify where total $= 4$; Dealing with both 2 2 1 1 situations and describing or showing 4's; $\underline{2\ 2}\ 1\ 1,\ 1\ \underline{2\ 2}\ 1$ or $1\ 1\ \underline{2\ 2}$; $\underline{2}\ 1\ 1\ 2,\ 1\ \underline{2}\ 1\ 2$ or $2\ 1\ \underline{2}\ 1$ |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4!\div 2! = 12$ arrangements of 0, 0, 1, 2 | B1 | Or $^4C_2 \times 2 = 12$ or write out these 12 |
| 4 arrangements of 0, 1, 1, 1 | B1 | Or write out these 4 |
| 4 arrangements of 0, 0, 0, 3 | B1 [3] | Or write out these 4 |
---2 Mo eats exactly 6 doughnuts in 4 days.\\
(i) What does the pigeonhole principle tell you about the number of doughnuts Mo eats in a day?
Mo eats exactly 6 doughnuts in 4 days, eating at least 1 doughnut each day.\\
(ii) Show that there must be either two consecutive days or three consecutive days on which Mo eats a total of exactly 4 doughnuts.
Mo eats exactly 3 identical jam doughnuts and exactly 3 identical iced doughnuts over the 4 days.\\
The number of jam doughnuts eaten on the four days is recorded as a list, for example $1,0,2,0$. The number of iced doughnuts eaten is not recorded.\\
(iii) Show that 20 different such lists are possible.
\hfill \mbox{\textit{OCR Further Discrete AS 2018 Q2 [7]}}