Question

There are 15 books on a shelf, 5 of which are on fiction. If five books are selected at random, find the probability that 4 of them are books on fiction.

• A. 5/72
• B. 5/273
• C. 50/3003
• D. 50/273
• E. None of these

Answer 1

The Correct Option is C

Step 1: Understand the problem.
There are 15 books on a shelf, and 5 of them are fiction books. We need to find the probability that 4 out of 5 randomly selected books are fiction books.

Step 2: Calculate the total number of ways to select 5 books from 15.
The total number of ways to select 5 books out of 15 is given by the combination formula:
\( \binom{15}{5} = \frac{15!}{5!(15-5)!} = \frac{15 \times 14 \times 13 \times 12 \times 11}{5 \times 4 \times 3 \times 2 \times 1} = 3003 \)

Step 3: Calculate the number of favorable outcomes.
To have 4 fiction books, we need to select 4 fiction books from the 5 available fiction books and 1 non-fiction book from the remaining 10 non-fiction books.
The number of ways to select 4 fiction books from 5 is:
\( \binom{5}{4} = 5 \)
The number of ways to select 1 non-fiction book from 10 is:
\( \binom{10}{1} = 10 \)
Therefore, the total number of favorable outcomes is:
\( \binom{5}{4} \times \binom{10}{1} = 5 \times 10 = 50 \)

Step 4: Calculate the probability.
The probability is the ratio of favorable outcomes to the total number of outcomes:
\( P(\text{4 fiction books}) = \frac{50}{3003} \)

Step 5: Conclusion.
The probability that 4 out of the 5 randomly selected books are fiction is \( \frac{50}{3003} \).

Final Answer:
The correct answer is \( \frac{50}{3003} \).