CAIE S1 2020 June — Question 4 4 marks

Exam BoardCAIE
ModuleS1 (Statistics 1)
Year2020
SessionJune
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicCombinations & Selection
TypeCommittee with gender/category constraints
DifficultyStandard +0.8 This requires systematic case analysis with multiple constraints (at least 2 pianists, at least 1 guitarist, more violinists than guitarists), careful enumeration of valid combinations, and multiple combination calculations. The constraint 'more violinists than guitarists' adds significant complexity beyond standard selection problems, requiring organized case-work and careful counting.
Spec5.01a Permutations and combinations: evaluate probabilities
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final. How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?
Question 4:
AnswerMarks Guidance
AnswerMark Guidance
Scenarios: 2P 3V 2G: \({}^8C_2 \times {}^4C_2 \times {}^6C_3 = 28\times6\times20 = 3360\); 2P 4V 1G: \({}^8C_2 \times {}^4C_1 \times {}^6C_4 = 28\times4\times15 = 1680\); 3P 3V 1G: \({}^8C_3 \times {}^4C_1 \times {}^6C_3 = 56\times4\times20 = 4480\); 4P 2V 1G: \({}^8C_4 \times {}^4C_1 \times {}^6C_2 = 70\times4\times15 = 4200\)M1 M1 for \({}^8C_r \times {}^4C_r \times {}^6C_r\) with \(\sum r = 7\)
Two unsimplified products correctB1
Summing the number of ways for 3 or 4 correct scenariosM1
Total: \(13720\)A1 
## Question 4:

| Answer | Mark | Guidance |
|--------|------|----------|
| Scenarios: 2P 3V 2G: ${}^8C_2 \times {}^4C_2 \times {}^6C_3 = 28\times6\times20 = 3360$; 2P 4V 1G: ${}^8C_2 \times {}^4C_1 \times {}^6C_4 = 28\times4\times15 = 1680$; 3P 3V 1G: ${}^8C_3 \times {}^4C_1 \times {}^6C_3 = 56\times4\times20 = 4480$; 4P 2V 1G: ${}^8C_4 \times {}^4C_1 \times {}^6C_2 = 70\times4\times15 = 4200$ | M1 | M1 for ${}^8C_r \times {}^4C_r \times {}^6C_r$ with $\sum r = 7$ |
| Two unsimplified products correct | B1 | |
| Summing the number of ways for 3 or 4 correct scenarios | M1 | |
| Total: $13720$ | A1 | |
4 In a music competition, there are 8 pianists, 4 guitarists and 6 violinists. 7 of these musicians will be selected to go through to the final.

How many different selections of 7 finalists can be made if there must be at least 2 pianists, at least 1 guitarist and more violinists than guitarists?\\

\hfill \mbox{\textit{CAIE S1 2020 Q4 [4]}}
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