| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Arrangements in multiple rows/groups |
| Difficulty | Standard +0.3 This is a straightforward combinations question with standard techniques: part (a) uses basic selection with division by overcounting (though boats are distinguishable by size, so it's just 6C3 × 3C2 × 1C1 = 60), and part (b) applies permutations with repetition and complementary counting. All methods are textbook exercises requiring no novel insight, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| \(^6C_3 \times\ ^3C_2 \times\ ^1C_1\) | M1 | \(^6C_a \times\ ^{6-a}C_b \times\ ^{6-a-b}C_{6-a-b}\) seen; \(^{6-a-b}C_{6-a-b}\) can be implied by 1 or omission, condone use of permutations |
| \(= 20 \times 3\) | A1 | Any correct method seen, no addition/additional scenarios |
| \(= 60\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{^6P_6}{^3P_3 \times\ ^2P_2 \times\ ^1P_1} = \dfrac{6!}{3\times2!}\) | M1 | \(^6P_6 / (^nP_n \times k)\) with \(3 \geq n > 1\) and \(6 \geq k\) an integer \(\geq 1\), not \(6!/1\) |
| \(= 60\) | A1 | Correct method with no additional terms; Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{4!}{3!} \times \dfrac{3!}{2!} \times 2\) | M1 | A single expression with either \(4!/3! \times k\) or \(3!/2! \times k\), \(k\) a positive integer seen (condone 2 identical expressions being added) |
| M1 | Correctly multiplying *their* single expression by 2 or 2 identical expressions being added | |
| \(= 24\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Total no. of arrangements \(= \dfrac{7!}{2!3!} = 420\ (A)\) | B1 | Accept unsimplified |
| No with 2s together \(= \dfrac{6!}{3!} = 120\ (B)\) | B1 | Accept unsimplified |
| With 2s not together: \(their\ (A) - their\ (B)\) | M1 | Subtraction indicated, possibly by *their* answer, no additional terms present |
| \(= 300\) ways | A1 | Exact value www |
| Answer | Marks | Guidance |
|---|---|---|
| \(\dfrac{5!}{3!} \times \dfrac{6\times5}{2}\) | B1 | \(k \times 5!\) in numerator, \(k\) a positive integer |
| B1 | \(m \times 3!\) in denominator, \(m\) a positive integer | |
| M1 | *Their* \(5!/3!\) multiplied by \(^6C_2\) only (no additional terms) | |
| \(= 300\) ways | A1 | Exact value www |
## Question 7(a):
$^6C_3 \times\ ^3C_2 \times\ ^1C_1$ | M1 | $^6C_a \times\ ^{6-a}C_b \times\ ^{6-a-b}C_{6-a-b}$ seen; $^{6-a-b}C_{6-a-b}$ can be implied by 1 or omission, condone use of permutations
$= 20 \times 3$ | A1 | Any correct method seen, no addition/additional scenarios
$= 60$ | A1 | Correct answer
**Alternative method:**
$\dfrac{^6P_6}{^3P_3 \times\ ^2P_2 \times\ ^1P_1} = \dfrac{6!}{3\times2!}$ | M1 | $^6P_6 / (^nP_n \times k)$ with $3 \geq n > 1$ and $6 \geq k$ an integer $\geq 1$, not $6!/1$
$= 60$ | A1 | Correct method with no additional terms; Correct answer
**Total: 3 marks**
---
## Question 7(b)(i):
$\dfrac{4!}{3!} \times \dfrac{3!}{2!} \times 2$ | M1 | A single expression with either $4!/3! \times k$ or $3!/2! \times k$, $k$ a positive integer seen (condone 2 identical expressions being added)
| M1 | Correctly multiplying *their* single expression by 2 or 2 identical expressions being added
$= 24$ | A1 | Correct answer
**Total: 3 marks**
---
## Question 7(b)(ii):
Total no. of arrangements $= \dfrac{7!}{2!3!} = 420\ (A)$ | B1 | Accept unsimplified
No with 2s together $= \dfrac{6!}{3!} = 120\ (B)$ | B1 | Accept unsimplified
With 2s not together: $their\ (A) - their\ (B)$ | M1 | Subtraction indicated, possibly by *their* answer, no additional terms present
$= 300$ ways | A1 | Exact value www
**Alternative method:**
$3\_7\_7\_7\_8\_$
$\dfrac{5!}{3!} \times \dfrac{6\times5}{2}$ | B1 | $k \times 5!$ in numerator, $k$ a positive integer
| B1 | $m \times 3!$ in denominator, $m$ a positive integer
| M1 | *Their* $5!/3!$ multiplied by $^6C_2$ only (no additional terms)
$= 300$ ways | A1 | Exact value www
**Total: 4 marks**7
\begin{enumerate}[label=(\alph*)]
\item A group of 6 teenagers go boating. There are three boats available. One boat has room for 3 people, one has room for 2 people and one has room for 1 person. Find the number of different ways the group of 6 teenagers can be divided between the three boats.
\item Find the number of different 7-digit numbers which can be formed from the seven digits 2, 2, 3, 7, 7, 7, 8 in each of the following cases.
\begin{enumerate}[label=(\roman*)]
\item The odd digits are together and the even digits are together.
\item The 2 s are not together.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2019 Q7 [10]}}