| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Digit arrangements forming numbers |
| Difficulty | Moderate -0.8 This is a straightforward permutations and combinations question with standard restrictions. Part (a) involves basic counting principles with simple constraints (digit position and parity), while part (b) uses standard combination formulas with case-by-case analysis. All techniques are routine for S1 level with no novel problem-solving required. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: \(3^{}, 4^{}, 6^{}, 8^{}\) | (M1 | \({}^5P_2\) or \({}^5C_2 \times 2!\) or \(5 \times 4\) OE (considering final 2 digits) |
| options \(4 \times 5 \times 4 = 80\) | M1 | Multiply by 4 or summing 4 options (considering first digit) |
| A1) | Correct final answer | |
| OR: Total number of values: \(6 \times 5 \times 4 = 120\) | (M1 | Calculating total number of values (with subtraction seen) |
| Number of values less than 300: \(2 \times 5 \times 4 = 40\) | M1 | Calculating number of unwanted values |
| Number of evens \(= 120 - 40 = 80\) | A1) | Correct final answer |
| 3 total |
| Answer | Marks | Guidance |
|---|---|---|
| \(3^{}, 4^{}, 6^{}, 8^{}\), options \(4 \times 6 \times 4\) (last) | (M1) | 6 linked to considering middle digit e.g. multiplied or in list |
| Multiply an integer by \(4 \times 4\) (condone \(\times 16\)) | M1 | No additional figures present for both M's to be awarded |
| \(= 96\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Total number of values \(4 \times 6 \times 6 = 144\) | (M1) | Calculating total number of values (with subtraction seen) |
| Number of odd values \(4 \times 6 \times 2 = 48\) | M1 | Calculating number of unwanted values |
| Number of evens \(= 144 - 48 = 96\) | A1 |
| Answer | Marks |
|---|---|
| \(252\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| B\((6)\)G\((4)\) | ||
| \(5\) in \(^6C_5 \times (x^4C_0) = 6 \times 1 = 6\) | M1 | Multiplying 2 combinations \(^6C_q \times ^4C_r\), \(q + r = 5\), or \(^6C_5\) seen alone |
| \(4\) in \(^6C_4 \times ^4C_1 = 15 \times 4 = 60\) | ||
| \(3\) in \(^6C_3 \times ^4C_2 = 20 \times 6 = 120\) | M1 | Summing 2 or 3 appropriate outcomes, involving perm/comb, no extra outcomes |
| Total \(= 186\) ways | A1 |
## Question 6(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| EITHER: $3^{**}, 4^{**}, 6^{**}, 8^{**}$ | (M1 | ${}^5P_2$ or ${}^5C_2 \times 2!$ or $5 \times 4$ OE (considering final 2 digits) |
| options $4 \times 5 \times 4 = 80$ | M1 | Multiply by 4 or summing 4 options (considering first digit) |
| | A1) | Correct final answer |
| OR: Total number of values: $6 \times 5 \times 4 = 120$ | (M1 | Calculating total number of values (with subtraction seen) |
| Number of values less than 300: $2 \times 5 \times 4 = 40$ | M1 | Calculating number of unwanted values |
| Number of evens $= 120 - 40 = 80$ | A1) | Correct final answer |
| | **3 total** | |
## Question 6(a)(ii):
**EITHER method:**
$3^{**}, 4^{**}, 6^{**}, 8^{**}$, options $4 \times 6 \times 4$ (last) | (M1) | 6 linked to considering middle digit e.g. multiplied or in list
Multiply an integer by $4 \times 4$ (condone $\times 16$) | M1 | No additional figures present for both M's to be awarded
$= 96$ | A1 |
**OR method:**
Total number of values $4 \times 6 \times 6 = 144$ | (M1) | Calculating total number of values (with subtraction seen)
Number of odd values $4 \times 6 \times 2 = 48$ | M1 | Calculating number of unwanted values
Number of evens $= 144 - 48 = 96$ | A1 |
**Total: 3 marks**
---
## Question 6(b)(i):
$252$ | B1 |
**Total: 1 mark**
---
## Question 6(b)(ii):
B$(6)$G$(4)$ | |
$5$ in $^6C_5 \times (x^4C_0) = 6 \times 1 = 6$ | M1 | Multiplying 2 combinations $^6C_q \times ^4C_r$, $q + r = 5$, or $^6C_5$ seen alone
$4$ in $^6C_4 \times ^4C_1 = 15 \times 4 = 60$ | |
$3$ in $^6C_3 \times ^4C_2 = 20 \times 6 = 120$ | M1 | Summing 2 or 3 appropriate outcomes, involving perm/comb, no extra outcomes
Total $= 186$ ways | A1 |
**Total: 3 marks**
---6
\begin{enumerate}[label=(\alph*)]
\item Find the number of different 3-digit numbers greater than 300 that can be made from the digits $1,2,3,4,6,8$ if
\begin{enumerate}[label=(\roman*)]
\item no digit can be repeated,
\item a digit can be repeated and the number made is even.
\end{enumerate}\item A team of 5 is chosen from 6 boys and 4 girls. Find the number of ways the team can be chosen if
\begin{enumerate}[label=(\roman*)]
\item there are no restrictions,
\item the team contains more boys than girls.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2017 Q6 [10]}}