In how many ways can a person read 3 different books from a collection of 7 books during a one-week vacation?

Note: [tex]${ }_n P _r=\frac{n!}{(n-r)!}$[/tex]



Answer :

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.