| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Permutations & Arrangements |
| Type | People arrangements in groups/rows |
| Difficulty | Moderate -0.3 This is a multi-part combinatorics question requiring systematic case-work for part (i) and straightforward application of permutation principles for parts (ii) and (iii). While it requires careful organization and multiple calculation steps, the techniques are standard A-level fare (combinations with constraints, arrangements with restrictions) with no novel problem-solving insight needed. Slightly easier than average due to clear structure and routine methods. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| W=1, S=1: \(3 = 6\times4\times {}^3C_3 = 24\) | M1 | Listing at least 4 different options |
| W=1, S=3: \(1 = 6\times {}^4C_3\times3 = 72\) | M1 | Mult 3 (combs) together assume \(6 = {}^6C_1\), \(\Sigma r = 5\) |
| W=3, S=1: \(1 = {}^6C_3\times4\times3 = 240\) | M1 | Summing at least 4 different evaluated/unsimplified options \(>1\) |
| W=1, S=2: \(2 = 6\times {}^4C_2\times {}^3C_2 = 108\) | ||
| W=2, S=1: \(2 = {}^6C_2\times4\times {}^3C_2 = 180\) | ||
| W=2, S=2: \(1 = {}^6C_2\times {}^4C_2\times3 = 270\) | B1 | At least 3 correct unsimplified options |
| Total \(= 894\) | A1 [5] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \({}^3P_2 \times {}^{10}P_8\) | B1 | \({}^3P_2\) oe seen multiplied either here or in (iii) |
| B1 | \(k^{10}P_x\) seen or \(k^y P_8\) with no addition, \(k \geq 1\), \(y > 8\), \(x < 10\) | |
| \(= 10886400\) | B1 [3] | Correct answer, nfww |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| DSWSWSWSWD or DWSWSWSWSD | If \({}^3P_2\) has not gained credit in (ii) may be awarded | |
| D in \({}^3P_2\) ways \(= 6\) | B1 | \({}^4P_4\) or \({}^6P_4\) oe seen multiplied or common in all terms (no division) |
| S in \({}^4P_4\) ways \(= 24\); W in \({}^6P_4 = 360\) | ||
| Swap SW in 2 ways | B1 | Mult by 2 (condone 2!) |
| Total \(= 103680\) ways | B1 [3] | Correct answer, 3sf or better, nfww |
## Question 7:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| W=1, S=1: $3 = 6\times4\times {}^3C_3 = 24$ | M1 | Listing at least 4 different options |
| W=1, S=3: $1 = 6\times {}^4C_3\times3 = 72$ | M1 | Mult 3 (combs) together assume $6 = {}^6C_1$, $\Sigma r = 5$ |
| W=3, S=1: $1 = {}^6C_3\times4\times3 = 240$ | M1 | Summing at least 4 different evaluated/unsimplified options $>1$ |
| W=1, S=2: $2 = 6\times {}^4C_2\times {}^3C_2 = 108$ | | |
| W=2, S=1: $2 = {}^6C_2\times4\times {}^3C_2 = 180$ | | |
| W=2, S=2: $1 = {}^6C_2\times {}^4C_2\times3 = 270$ | B1 | At least 3 correct unsimplified options |
| Total $= 894$ | A1 **[5]** | Correct answer |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| ${}^3P_2 \times {}^{10}P_8$ | B1 | ${}^3P_2$ oe seen multiplied either here or in (iii) |
| | B1 | $k^{10}P_x$ seen or $k^y P_8$ with no addition, $k \geq 1$, $y > 8$, $x < 10$ |
| $= 10886400$ | B1 **[3]** | Correct answer, nfww |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| DSWSWSWSWD or DWSWSWSWSD | | If ${}^3P_2$ has not gained credit in (ii) may be awarded |
| D in ${}^3P_2$ ways $= 6$ | B1 | ${}^4P_4$ or ${}^6P_4$ oe seen multiplied or common in all terms (no division) |
| S in ${}^4P_4$ ways $= 24$; W in ${}^6P_4 = 360$ | | |
| Swap SW in 2 ways | B1 | Mult by 2 (condone 2!) |
| Total $= 103680$ ways | B1 **[3]** | Correct answer, 3sf or better, nfww |7 Rachel has 3 types of ornament. She has 6 different wooden animals, 4 different sea-shells and 3 different pottery ducks.\\
(i) She lets her daughter Cherry choose 5 ornaments to play with. Cherry chooses at least 1 of each type of ornament. How many different selections can Cherry make?
Rachel displays 10 of the 13 ornaments in a row on her window-sill. Find the number of different arrangements that are possible if\\
(ii) she has a duck at each end of the row and no ducks anywhere else,\\
(iii) she has a duck at each end of the row and wooden animals and sea-shells are placed alternately in the positions in between.
\hfill \mbox{\textit{CAIE S1 2015 Q7 [11]}}