To determine the number of ways a person can read 3 different books from a collection of 7 books during a one-week vacation, we need to calculate the number of permutations. Permutations are used when the order in which items are chosen matters.
The formula for permutations is given by:
[tex]\[{}_n P_r = \frac{n!}{(n-r)!}\][/tex]
In this problem:
- [tex]\(n\)[/tex] is the total number of books, which is 7.
- [tex]\(r\)[/tex] is the number of books to read, which is 3.
Applying these values to the formula, we have:
[tex]\[{}_7 P_3 = \frac{7!}{(7-3)!}\][/tex]
Next, we will calculate the factorials:
1. [tex]\(7!\)[/tex] (7 factorial) is the product of all positive integers up to 7.
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040
\][/tex]
2. [tex]\((7-3)!\)[/tex], which is [tex]\(4!\)[/tex]:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
Now, substitute these factorial values into the permutation formula:
[tex]\[
{}_7 P_3 = \frac{5040}{24}
\][/tex]
Perform the division:
[tex]\[
\frac{5040}{24} = 210
\][/tex]
Therefore, the number of ways a person can read 3 different books from a collection of 7 books during a one-week vacation is [tex]\(\boxed{210}\)[/tex] ways.
