| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Arranging identical items in a line |
| Difficulty | Standard +0.3 This is a multi-part combinatorics question requiring standard techniques: (i) uses basic counting with odd number constraint, (ii) involves arranging items with restrictions (gap method), and (iii) uses permutations of identical objects with a grouping constraint. While it requires multiple techniques across three parts, each individual part follows textbook methods without requiring novel insight or particularly complex reasoning. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.01b Selection/arrangement: probability problems |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(^{12}C_1 + ^{12}C_3 + ^{12}C_5 + ^{12}C_7 + ^{12}C_9 + ^{12}C_{11}\) | M1 | Summing at least 4 \(^{12}C_x\) combinations with \(x\) = odd numbers |
| A1 | Correct unsimplified answer (can be implied by final answer) | |
| \(= 2048\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(7!\times\, ^8P_4\) | B1 | \(7!\) seen alone or multiplied only (cupcakes ordered) |
| M1 | Multiplying by \(^8P_4\) o.e. (placing brownies) | |
| \(= 8467200\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9!\,/\,(6!\times 2!)\) | B1 | \(9!\) o.e. seen alone or as numerator |
| M1 | Dividing by at least one of \(6!, 2!\) (removing repeated shortbread or gingerbread biscuits); ignore \(4!\) if present | |
| \(= 252\) | A1 | Correct answer |
# Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $^{12}C_1 + ^{12}C_3 + ^{12}C_5 + ^{12}C_7 + ^{12}C_9 + ^{12}C_{11}$ | M1 | Summing at least 4 $^{12}C_x$ combinations with $x$ = odd numbers |
| | A1 | Correct unsimplified answer (can be implied by final answer) |
| $= 2048$ | A1 | Correct answer |
---
# Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $7!\times\, ^8P_4$ | B1 | $7!$ seen alone or multiplied only (cupcakes ordered) |
| | M1 | Multiplying by $^8P_4$ o.e. (placing brownies) |
| $= 8467200$ | A1 | Correct answer |
---
# Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9!\,/\,(6!\times 2!)$ | B1 | $9!$ o.e. seen alone or as numerator |
| | M1 | Dividing by at least one of $6!, 2!$ (removing repeated shortbread or gingerbread biscuits); ignore $4!$ if present |
| $= 252$ | A1 | Correct answer |5 (i) A plate of cakes holds 12 different cakes. Find the number of ways these cakes can be shared between Alex and James if each receives an odd number of cakes.\\
(ii) Another plate holds 7 cup cakes, each with a different colour icing, and 4 brownies, each of a different size. Find the number of different ways these 11 cakes can be arranged in a row if no brownie is next to another brownie.\\
(iii) A plate of biscuits holds 4 identical chocolate biscuits, 6 identical shortbread biscuits and 2 identical gingerbread biscuits. These biscuits are all placed in a row. Find how many different arrangements are possible if the chocolate biscuits are all kept together.\\
\hfill \mbox{\textit{CAIE S1 2017 Q5 [9]}}