| Exam Board | OCR |
|---|---|
| Module | Further Discrete AS (Further Discrete AS) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Counting with digit/number constraints |
| Difficulty | Moderate -0.8 This question tests basic combinatorics concepts (combinations vs permutations) and understanding of problem types, but requires minimal calculation and no complex problem-solving. Parts (a)-(c) are straightforward applications of counting principles with simple explanations, while part (d) tests definition recall. The most demanding aspect is articulating clear explanations, but the mathematical content is elementary for Further Maths students. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities7.01a Types of problem: existence, construction, enumeration, optimisation7.01f Combinations: unordered subsets of r from n elements7.01g Arrangements in a line: with repetition and restriction |
| Answer | Marks | Guidance |
|---|---|---|
| \(^{15}C_2 = 105\) | M1 (AO 1.1), A1 (AO 1.1) | \(^{15}C_2\) or equivalent seen or implied; If M0 scored then SC1 for 105 as final answer; if M0 scored SC1 for 210 as final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Some numbers can be made in more than one way, e.g. 1 and 11 give 111 both ways round | B1 (AO 2.4) | Any suitable example showing two different orders giving the same result; e.g. 1 followed by 13 or 11 followed by 3 both give 113 |
| Answer | Marks | Guidance |
|---|---|---|
| Lead digit is 1 when first card is 1 or any of 10, 11, 12, 13, 14, 15 | B1 (AO 2.4) | Identifying that double digit cards also start with 1; 7 of the cards have lead digit 1; Benford's law; BOD 6 or 8 |
| Answer | Marks | Guidance |
|---|---|---|
| He does not need to actually find these primes, only find out how many there are | B1 (AO 2.4) | It is an enumeration problem; Constructing some solutions would not necessarily give a full count |
# Question 1:
## Part (a)
$^{15}C_2 = 105$ | M1 (AO 1.1), A1 (AO 1.1) | $^{15}C_2$ or equivalent seen or implied; If M0 scored then SC1 for 105 as final answer; if M0 scored SC1 for 210 as final answer
## Part (b)
Some numbers can be made in more than one way, e.g. 1 and 11 give 111 both ways round | B1 (AO 2.4) | Any suitable example showing two different orders giving the same result; e.g. 1 followed by 13 or 11 followed by 3 both give 113
## Part (c)
Lead digit is 1 when first card is 1 or any of 10, 11, 12, 13, 14, 15 | B1 (AO 2.4) | Identifying that double digit cards also start with 1; 7 of the cards have lead digit 1; Benford's law; BOD 6 or 8
## Part (d)
He does not need to actually find these primes, only find out how many there are | B1 (AO 2.4) | It is an enumeration problem; Constructing some solutions would not necessarily give a full count
**Total: [5]**
---1 Alfie has a set of 15 cards numbered consecutively from 1 to 15.\\
He chooses two of the cards.
\begin{enumerate}[label=(\alph*)]
\item How many different sets of two cards are possible?
Alfie places the two cards side by side to form a number with 2,3 or 4 digits.
\item Explain why there are fewer than ${ } ^ { 15 } \mathrm { P } _ { 2 } = 210$ possible numbers that can be made.
\item Explain why, with these cards, 1 is the lead digit more often than any other digit.
Alfie makes the number 113, which is a 3-digit prime number.
Alfie says that the problem of working out how many 3-digit prime numbers can be made using two of the cards is a construction problem, because he is trying to find all of them.
\item Explain why Alfie is wrong to say this is a construction problem.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Discrete AS 2019 Q1 [5]}}