Answer :
Okay, let's go through each of the statements one by one to determine if they are true or not.
1. There are [tex]\({}_{20}C_3\)[/tex] possible ways to choose three books to read.
To find the number of ways to choose 3 books out of a total of 5 mysteries + 7 biographies + 8 science fiction novels, we use the combination formula [tex]\( \binom{n}{k} \)[/tex]:
[tex]\[ \binom{20}{3} \][/tex]
We know from our results that [tex]\(\binom{20}{3} = 1140\)[/tex]. Therefore, this statement is true.
2. There are [tex]\({}_{5}C_3\)[/tex] possible ways to choose three mysteries to read.
To find the number of ways to choose 3 mysteries out of 5, we again use the combination formula [tex]\( \binom{n}{k} \)[/tex]:
[tex]\[ \binom{5}{3} \][/tex]
We know from our results that [tex]\(\binom{5}{3} = 10\)[/tex]. Therefore, this statement is true.
3. There are [tex]\({}_{15}C_3\)[/tex] possible ways to choose three books that are not all mysteries.
This statement is implying considering the choice of books excluding the mysteries. However, the statement should be evaluated correctly. According to the results:
[tex]\[ \binom{15}{3} = 455 \][/tex]
This indicates that there are 455 ways to choose three books among the 15 remaining books (excluding the mysteries). So, this statement is true.
4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{}_{5}C_3}\)[/tex].
This means that the probability of choosing 3 mysteries from 5 mysteries is:
[tex]\[ \frac{1}{\binom{5}{3}} \][/tex]
We know from our results that the probability is [tex]\(0.1\)[/tex], which is [tex]\(\frac{1}{10}\)[/tex]. Given that [tex]\(\binom{5}{3} = 10\)[/tex], the expression [tex]\(\frac{1}{\binom{5}{3}}\)[/tex] is indeed correct. Therefore, this statement is true.
5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{}_{5}C_3}{20C_3}\)[/tex].
This is the probability complement rule where the probability of not choosing all mysteries is 1 minus the probability of choosing all mysteries:
[tex]\[ 1 - \frac{\binom{5}{3}}{\binom{20}{3}} \][/tex]
We know from our results that:
[tex]\[ 1 - \frac{10}{1140} = 0.9912280701754386 \][/tex]
This confirms the statement is true.
All things considered, the statements that are true are:
1. [tex]\({}_{20}C_3\)[/tex] possible ways to choose three books to read.
2. [tex]\({}_{5}C_3\)[/tex] possible ways to choose three mysteries to read.
3. [tex]\({}_{15}C_3\)[/tex] possible ways to choose three books that are not all mysteries.
4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{}_{5}C_3}\)[/tex].
5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{}_{5}C_3}{20C_3}\)[/tex].
Thus, all the given statements are true.
1. There are [tex]\({}_{20}C_3\)[/tex] possible ways to choose three books to read.
To find the number of ways to choose 3 books out of a total of 5 mysteries + 7 biographies + 8 science fiction novels, we use the combination formula [tex]\( \binom{n}{k} \)[/tex]:
[tex]\[ \binom{20}{3} \][/tex]
We know from our results that [tex]\(\binom{20}{3} = 1140\)[/tex]. Therefore, this statement is true.
2. There are [tex]\({}_{5}C_3\)[/tex] possible ways to choose three mysteries to read.
To find the number of ways to choose 3 mysteries out of 5, we again use the combination formula [tex]\( \binom{n}{k} \)[/tex]:
[tex]\[ \binom{5}{3} \][/tex]
We know from our results that [tex]\(\binom{5}{3} = 10\)[/tex]. Therefore, this statement is true.
3. There are [tex]\({}_{15}C_3\)[/tex] possible ways to choose three books that are not all mysteries.
This statement is implying considering the choice of books excluding the mysteries. However, the statement should be evaluated correctly. According to the results:
[tex]\[ \binom{15}{3} = 455 \][/tex]
This indicates that there are 455 ways to choose three books among the 15 remaining books (excluding the mysteries). So, this statement is true.
4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{}_{5}C_3}\)[/tex].
This means that the probability of choosing 3 mysteries from 5 mysteries is:
[tex]\[ \frac{1}{\binom{5}{3}} \][/tex]
We know from our results that the probability is [tex]\(0.1\)[/tex], which is [tex]\(\frac{1}{10}\)[/tex]. Given that [tex]\(\binom{5}{3} = 10\)[/tex], the expression [tex]\(\frac{1}{\binom{5}{3}}\)[/tex] is indeed correct. Therefore, this statement is true.
5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{}_{5}C_3}{20C_3}\)[/tex].
This is the probability complement rule where the probability of not choosing all mysteries is 1 minus the probability of choosing all mysteries:
[tex]\[ 1 - \frac{\binom{5}{3}}{\binom{20}{3}} \][/tex]
We know from our results that:
[tex]\[ 1 - \frac{10}{1140} = 0.9912280701754386 \][/tex]
This confirms the statement is true.
All things considered, the statements that are true are:
1. [tex]\({}_{20}C_3\)[/tex] possible ways to choose three books to read.
2. [tex]\({}_{5}C_3\)[/tex] possible ways to choose three mysteries to read.
3. [tex]\({}_{15}C_3\)[/tex] possible ways to choose three books that are not all mysteries.
4. The probability that Mariah will choose 3 mysteries can be expressed as [tex]\(\frac{1}{{}_{5}C_3}\)[/tex].
5. The probability that Mariah will not choose all mysteries can be expressed as [tex]\(1 - \frac{{}_{5}C_3}{20C_3}\)[/tex].
Thus, all the given statements are true.
