| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | At least/at most N letters between items |
| Difficulty | Challenging +1.2 Part (a) is a standard permutation with repeated letters (9!/2!). Part (b) requires complementary counting or casework to handle the constraint on positions of the two As, which is a step beyond routine but still a well-practiced technique in S1. The multi-step nature and need to systematically enumerate cases (0, 1, or 2 letters between As) elevates this above average difficulty. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{9!}{2!} = 181440\) | B1 | Exact value must be seen. CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Scenario: AA \([\wedge\wedge\wedge\wedge\wedge\wedge\wedge]\): \(7! \times 8\) or \(8!\) \([= 40320]\) | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified |
| \(A\wedge A[\wedge\wedge\wedge\wedge\wedge\wedge]\): \(7! \times 7\) \([= 35280]\); \(A\wedge\wedge A[\wedge\wedge\wedge\wedge\wedge]\): \(7! \times 6\) \([= 30240]\); Total: \(7! \times (8+7+6)\) | M1 | Add values of 3 correct scenarios, no incorrect/repeated scenarios |
| \(= 105\,840\) | A1 | If M1 not awarded, SC B1 for 105840 WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| AA \([\wedge\wedge\wedge\wedge\wedge\wedge\wedge]\): \(8!\) \([= 40320]\) | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified |
| \(A\wedge A[\wedge\wedge\wedge\wedge\wedge\wedge]\): \({}^7P_1 \times 7!\) or \({}^7C_1 \times 7!\) \([= 35280]\); \(A\wedge\wedge A[\wedge\wedge\wedge\wedge\wedge]\): \({}^7P_2 \times 6!\) or \({}^7C_2 \times 2 \times 6!\) \([= 30240]\); Total: \(8! + {}^7P_1 \times 7! + {}^7P_2 \times 6!\) or \(8! + {}^7C_1 \times 7! + {}^7C_2 \times 2 \times 6!\) | M1 | Add values of 3 correct scenarios, no incorrect/repeated scenarios |
| \(= 105\,840\) | A1 | If M1 not awarded, SC B1 for 105840 WWW |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(A\wedge\wedge\wedge A[\wedge\wedge\wedge\wedge]\): \(7!\times 5\) \([=25200]\); \(A\wedge\wedge\wedge\wedge A[\wedge\wedge\wedge]\): \(7!\times 4\) \([=20160]\); \(A\wedge\wedge\wedge\wedge\wedge A[\wedge\wedge]\): \(7!\times 3\) \([=15120]\); \(A\wedge\wedge\wedge\wedge\wedge\wedge A[\wedge]\): \(7!\times 2\) \([=10080]\); \(A\wedge\wedge\wedge\wedge\wedge\wedge\wedge A\): \(7![\times 1]\) \([=5040]\) | B1 | Correct outcome/value for 1 identified scenario |
| Total \(= \frac{9!}{2!} - 7!\times(5+4+3+2+1)\) | M1 | Their 2(a), or correct, subtract values of 5 correct scenarios, no incorrect/repeated scenarios |
| \(= 105\,840\) | A1 | If M1 not awarded, SC B1 for 105840 WWW |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{9!}{2!} = 181440$ | B1 | Exact value must be seen. CAO |
**Total: 1 mark**
---
## Question 2(b):
**Method 1**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Scenario: AA $[\wedge\wedge\wedge\wedge\wedge\wedge\wedge]$: $7! \times 8$ or $8!$ $[= 40320]$ | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified |
| $A\wedge A[\wedge\wedge\wedge\wedge\wedge\wedge]$: $7! \times 7$ $[= 35280]$; $A\wedge\wedge A[\wedge\wedge\wedge\wedge\wedge]$: $7! \times 6$ $[= 30240]$; Total: $7! \times (8+7+6)$ | M1 | Add values of 3 correct scenarios, no incorrect/repeated scenarios |
| $= 105\,840$ | A1 | If M1 not awarded, SC B1 for 105840 WWW |
**Method 2**
| Answer | Marks | Guidance |
|--------|-------|----------|
| AA $[\wedge\wedge\wedge\wedge\wedge\wedge\wedge]$: $8!$ $[= 40320]$ | B1 | Correct outcome/value for 1 identified scenario, accept unsimplified |
| $A\wedge A[\wedge\wedge\wedge\wedge\wedge\wedge]$: ${}^7P_1 \times 7!$ or ${}^7C_1 \times 7!$ $[= 35280]$; $A\wedge\wedge A[\wedge\wedge\wedge\wedge\wedge]$: ${}^7P_2 \times 6!$ or ${}^7C_2 \times 2 \times 6!$ $[= 30240]$; Total: $8! + {}^7P_1 \times 7! + {}^7P_2 \times 6!$ or $8! + {}^7C_1 \times 7! + {}^7C_2 \times 2 \times 6!$ | M1 | Add values of 3 correct scenarios, no incorrect/repeated scenarios |
| $= 105\,840$ | A1 | If M1 not awarded, SC B1 for 105840 WWW |
**Method 3**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $A\wedge\wedge\wedge A[\wedge\wedge\wedge\wedge]$: $7!\times 5$ $[=25200]$; $A\wedge\wedge\wedge\wedge A[\wedge\wedge\wedge]$: $7!\times 4$ $[=20160]$; $A\wedge\wedge\wedge\wedge\wedge A[\wedge\wedge]$: $7!\times 3$ $[=15120]$; $A\wedge\wedge\wedge\wedge\wedge\wedge A[\wedge]$: $7!\times 2$ $[=10080]$; $A\wedge\wedge\wedge\wedge\wedge\wedge\wedge A$: $7![\times 1]$ $[=5040]$ | B1 | Correct outcome/value for 1 identified scenario |
| Total $= \frac{9!}{2!} - 7!\times(5+4+3+2+1)$ | M1 | Their 2(a), or correct, subtract values of 5 correct scenarios, no incorrect/repeated scenarios |
| $= 105\,840$ | A1 | If M1 not awarded, SC B1 for 105840 WWW |
**Total: 3 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Find the number of different arrangements of the 9 letters in the word ALGEBRAIC.
\item Find the number of different arrangements of the 9 letters in the word ALGEBRAIC in which there are no more than two letters between the two As.\\
\includegraphics[max width=\textwidth, alt={}, center]{aeb7b26e-6754-4c61-b71e-e8169c617b91-04_2718_38_107_2009}
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q2 [4]}}