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Answer:
a) 2002 different groups of 5 problems can be chosen from the 14 problems.
b) P=0.04995%
c) 21 different groups of 5.
P=1.04895%
Step-by-step explanation:
a)
In order to solve this problem we must ask ourselves: Does the order matter? We can see that in this case the order in which the problems are chosen doesn't really matter, so we are talking about a combination:
[tex]_{n}C_{r}=\frac{n!}{r!(n-r)!}[/tex]
in this case, n=14 while r=5, so our combination will be:
[tex]_{14}C_{5}=\frac{14!}{5!(14-5)!}[/tex]
which yields:
[tex]_{14}C_{5}=2002[/tex] which will be the answer for part a.
b)
In order to find the probability we can use the probability formula:
[tex]Probability=\frac{desired \hspace{5}outcome}{number\hspace{5} of \hspace{5} possible \hspace{5} outcomes}[/tex]
so in this case there is only 1 desired outcome out of the 2002 possible outcomes so our probability is:
[tex]P=\frac{1}{2002}[/tex]
which is the same as:
P=0.04995%
c) The same applies to part c. In order to find the number of groups Silvia did, we can make us of the combination formula:
[tex]_{7}C_{5}=\frac{7!}{5!(7-5)!}[/tex]
which yields:
[tex]_{14}C_{5}=21[/tex]
so there are 21 different groups of 5 she did by solving 7 problems.
So we can use the same formula to find the probability that one of the groups she completed comprised the group selected to be graded so we get:
[tex]P=\frac{21}{2002}=\frac{3}{286}[/tex]
which is the same as:
P=1.04895%
(remember that in order to turn a decimal number into a percentage you need to multiply the decimal number by 100)