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? ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ BLANK PAGE
Exam No：9709_s20_qp_51 Year：2020 Question No：4
Answer:
Scenarios:
$
\begin{array}{ll}
\text { 2P 3V 2G } & { }^{8} \mathrm{C}_{2} \times{ }^{4} \mathrm{C}_{2} \times{ }^{6} \mathrm{C}_{3}=28 \times 6 \times 20=3360 \\
\text { 2P 4V 1G } & { }^{8} \mathrm{C}_{2} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{4}=28 \times 4 \times 15=1680 \\
\text { 3P 3V 1G } & { }^{8} \mathrm{C}_{3} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{3}=56 \times 4 \times 20=4480 \\
\text { 4P 2V 1G } & { }^{8} \mathrm{C}_{4} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{2}=70 \times 4 \times 15=4200 \\
\text { (M1 for }{ }^{8} \mathrm{C}_{\mathrm{r}} \times{ }^{4} \mathrm{C}_{\mathrm{r}} \times{ }^{6} \mathrm{C}_{\mathrm{r}} \text { with } \sum r=7 \text { ) }
\end{array}
$
Two unsimplified products correct
Summing the number of ways for 3 or 4 correct scenarios
Total: 13720
$
\begin{array}{ll}
\text { 2P 3V 2G } & { }^{8} \mathrm{C}_{2} \times{ }^{4} \mathrm{C}_{2} \times{ }^{6} \mathrm{C}_{3}=28 \times 6 \times 20=3360 \\
\text { 2P 4V 1G } & { }^{8} \mathrm{C}_{2} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{4}=28 \times 4 \times 15=1680 \\
\text { 3P 3V 1G } & { }^{8} \mathrm{C}_{3} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{3}=56 \times 4 \times 20=4480 \\
\text { 4P 2V 1G } & { }^{8} \mathrm{C}_{4} \times{ }^{4} \mathrm{C}_{1} \times{ }^{6} \mathrm{C}_{2}=70 \times 4 \times 15=4200 \\
\text { (M1 for }{ }^{8} \mathrm{C}_{\mathrm{r}} \times{ }^{4} \mathrm{C}_{\mathrm{r}} \times{ }^{6} \mathrm{C}_{\mathrm{r}} \text { with } \sum r=7 \text { ) }
\end{array}
$
Two unsimplified products correct
Summing the number of ways for 3 or 4 correct scenarios
Total: 13720
Knowledge points:
5.2.1 understand the terms permutation and combination, and solve simple problems involving selections
Solution:
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