| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Multi-stage selection problems |
| Difficulty | Standard +0.3 This is a standard combinations question with straightforward case-by-case counting. Part (a) requires one calculation: C(5,1)×C(7,4). Part (b) needs casework (2 boys + 1 girl, or 3 boys) but the cases are obvious. Part (c) combines permutations with restrictions using the 'treat as one unit' technique. All methods are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(^5C_1 \times ^7C_4\) | M1 | \(^7C_4 \times k\), \(k\) integer \(\geqslant 1\). Condone \(^5P_1\) for M1 only |
| \(175\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 2B 1G 2A: \(^3C_2 \times ^4C_1 \times ^5C_2 = 120\); 2B 2G 1A: \(^3C_2 \times ^4C_2 \times ^5C_1 = 90\); 2B 3G: \(^3C_2 \times ^4C_3 = 12\); 3B 1G 1A: \(^3C_3 \times ^4C_1 \times ^5C_1 = 20\); 3B 2G: \(^3C_3 \times ^4C_2 = 6\) | M1 | \(^3C_x \times ^4C_y \times ^5C_z\), \(x+y+z=5\), \(x,y,z\) integers \(\geqslant 1\). Condone use of permutations for this mark |
| (2 appropriate identified outcomes correct) | B1 | 2 appropriate identified outcomes correct, allow unsimplified |
| (summing their values for 4 or 5 correct identified scenarios only) | M1 | Summing *their* values for 4 or 5 correct identified scenarios only (no repeats or additional scenarios), condone identification by unsimplified expressions |
| \([\text{Total} =]\ 248\) | A1 | Note: Only dependent upon M marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(8! \times 3! \times ^5P_2\) | M1 | \(8! \times m\), \(m\) an integer \(\geqslant 1\). Accept \(8 \times 7!\) for \(8!\) |
| M1 | \(3! \times n\), \(n\) an integer \(> 1\) | |
| M1 | \(p \times ^5P_2\), \(p \times ^5C_2 \times 2\), \(p \times 20\), \(p\) an integer \(> 1\). If extra terms present, maximum 2/3 M marks available | |
| \(4838400\) | A1 | Exact value required |
## Question 5(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $^5C_1 \times ^7C_4$ | M1 | $^7C_4 \times k$, $k$ integer $\geqslant 1$. Condone $^5P_1$ for M1 only |
| $175$ | A1 | |
## Question 5(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 2B 1G 2A: $^3C_2 \times ^4C_1 \times ^5C_2 = 120$; 2B 2G 1A: $^3C_2 \times ^4C_2 \times ^5C_1 = 90$; 2B 3G: $^3C_2 \times ^4C_3 = 12$; 3B 1G 1A: $^3C_3 \times ^4C_1 \times ^5C_1 = 20$; 3B 2G: $^3C_3 \times ^4C_2 = 6$ | M1 | $^3C_x \times ^4C_y \times ^5C_z$, $x+y+z=5$, $x,y,z$ integers $\geqslant 1$. Condone use of permutations for this mark |
| (2 appropriate identified outcomes correct) | B1 | 2 appropriate identified outcomes correct, allow unsimplified |
| (summing their values for 4 or 5 correct identified scenarios only) | M1 | Summing *their* values for 4 or 5 correct identified scenarios only (no repeats or additional scenarios), condone identification by unsimplified expressions |
| $[\text{Total} =]\ 248$ | A1 | Note: Only dependent upon M marks |
## Question 5(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $8! \times 3! \times ^5P_2$ | M1 | $8! \times m$, $m$ an integer $\geqslant 1$. Accept $8 \times 7!$ for $8!$ |
| | M1 | $3! \times n$, $n$ an integer $> 1$ |
| | M1 | $p \times ^5P_2$, $p \times ^5C_2 \times 2$, $p \times 20$, $p$ an integer $> 1$. If extra terms present, maximum 2/3 M marks available |
| $4838400$ | A1 | Exact value required |5 A group of 12 people consists of 3 boys, 4 girls and 5 adults.
\begin{enumerate}[label=(\alph*)]
\item In how many ways can a team of 5 people be chosen from the group if exactly one adult is included?
\item In how many ways can a team of 5 people be chosen from the group if the team includes at least 2 boys and at least 1 girl?\\
The same group of 12 people stand in a line.
\item How many different arrangements are there in which the 3 boys stand together and an adult is at each end of the line?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q5 [10]}}