Answer:
56/8671
Explanation:
First, determine the total number of artworks.
• Sculptures = 10
,• Sketches = 11
,• Oil paintings = 9
Total = 10+11+9 = 30
20 artworks can be selected out of 30 in 30C20 ways.
Next:
• 4 sculptures can be selected out of 10 in 10C4 ways.
,• 10 sketches can be selected out of 11 in 11C10 ways.
,• 6 oil paintings can be selected out of 9 in 9C6 ways.
The combination formula is:
[tex]^nC_x=\frac{n!}{(n-x)!x!}[/tex]Therefore:
[tex]\begin{gathered} ^{10}C_4=\frac{10!}{(10-4)!4!}=\frac{10!}{6!4!}=210 \\ ^{11}C_{10}=\frac{11!}{(11-10)!10!}=\frac{11!}{1!10!}=11 \\ ^9C_6=\frac{9!}{(9-6)!6!}=\frac{9!}{3!6!}=84 \\ ^{30}C_{20}=\frac{30!}{(30-20)!20!}=\frac{30!}{10!20!}=30045015 \end{gathered}[/tex]Thus, the probability that 4 sculptures, 10 sketches, and 6 oil paintings are chosen to be displayed is:
[tex]\begin{gathered} \frac{^{10}C_4\times^{11}C_{10}\times^9C_6}{^{30}C_{20}} \\ =\frac{210\times11\times84}{30045015} \\ =\frac{194040}{30045015} \\ =\frac{56}{8671} \end{gathered}[/tex]The probability is 56/8671.
