| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | Seating arrangements with constraints |
| Difficulty | Moderate -0.8 This is a straightforward permutations and combinations question with standard constraints. Part (a)(i) is basic permutations P(40,5), part (a)(ii) requires counting arrangements with simple adjacency constraints using standard techniques, and part (b) is a routine 'complementary counting' problem. All parts use textbook methods with no novel insight required, making this easier than average for A-level. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(^{40}P_5\) | M1 | \(^{40}P_x\) or \(^yP_5\) seen, can be mult by \(k \geq 1\) |
| \(= 78\,960\,960\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Not front row e.g. WEJ** in \(3 \times 3! = 18\) ways | B1 | \(3!\) seen mult by \(k \geq 1\) |
| 7 rows in \(7 \times 18 = 126\) ways | B1 | mult by 7 |
| Front row e.g. *MA** in \(4 \times 2 = 8\) ways | M1 | Attempt at front row arrangements and multiplying by the 7 other rows arrangements, need not be correct |
| Total \(126 \times 8 = 1008\) | A1 | |
| 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: e.g. \(^8C_3 = 56\) ways (R in), \(^8C_3 = 56\) ways (L in) | (M1 | Considering either R or L only in team |
| \(^8C_4 = 70\) ways (neither in team) | M1* | Considering neither in team |
| DM1 | Summing 3 scenarios | |
| Total 182 ways | A1) | |
| OR1: No restrictions \(^{10}C_4 = 210\) ways | (M1 | \(^{10}C_4 -\), considering no restrictions with subtraction |
| \(^8C_2 = 28\) (both in team) | M1* | Considering both in team |
| \(210 - 28 = 182\) ways | DM1, A1) | Subtract |
| OR2: R out \(^9C_4 = 126\), L out \(^9C_4 = 126\) | (M1 | Considering either R out or L out |
| Both out \(^8C_4 = 70\) | M1* | Considering both out |
| \(126 + 126 - 70 = 182\) ways | DM1, A1) | Summing 2 scenarios and subtracting 1 scenario |
| 4 |
## Question 6:
### Part 6(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $^{40}P_5$ | M1 | $^{40}P_x$ or $^yP_5$ seen, can be mult by $k \geq 1$ |
| $= 78\,960\,960$ | A1 | |
| | **2** | |
### Part 6(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Not front row e.g. WEJ** in $3 \times 3! = 18$ ways | B1 | $3!$ seen mult by $k \geq 1$ |
| 7 rows in $7 \times 18 = 126$ ways | B1 | mult by 7 |
| Front row e.g. *MA** in $4 \times 2 = 8$ ways | M1 | Attempt at front row arrangements and multiplying by the 7 other rows arrangements, need not be correct |
| Total $126 \times 8 = 1008$ | A1 | |
| | **4** | |
### Part 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** e.g. $^8C_3 = 56$ ways (R in), $^8C_3 = 56$ ways (L in) | (M1 | Considering either R or L only in team |
| $^8C_4 = 70$ ways (neither in team) | M1* | Considering neither in team |
| | DM1 | Summing 3 scenarios |
| Total 182 ways | A1) | |
| **OR1:** No restrictions $^{10}C_4 = 210$ ways | (M1 | $^{10}C_4 -$, considering no restrictions with subtraction |
| $^8C_2 = 28$ (both in team) | M1* | Considering both in team |
| $210 - 28 = 182$ ways | DM1, A1) | Subtract |
| **OR2:** R out $^9C_4 = 126$, L out $^9C_4 = 126$ | (M1 | Considering either R out or L out |
| Both out $^8C_4 = 70$ | M1* | Considering both out |
| $126 + 126 - 70 = 182$ ways | DM1, A1) | Summing 2 scenarios and subtracting 1 scenario |
| | **4** | |
---6
\begin{enumerate}[label=(\alph*)]
\item A village hall has seats for 40 people, consisting of 8 rows with 5 seats in each row. Mary, Ahmad, Wayne, Elsie and John are the first to arrive in the village hall and no seats are taken before they arrive.
\begin{enumerate}[label=(\roman*)]
\item How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John assuming there are no restrictions?
\item How many possible arrangements are there of seating Mary, Ahmad, Wayne, Elsie and John if Mary and Ahmad sit together in the front row and the other three sit together in one of the other rows?
\end{enumerate}\item In how many ways can a team of 4 people be chosen from 10 people if 2 of the people, Ross and Lionel, refuse to be in the team together?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2017 Q6 [10]}}