There are 20 books on a summer reading list. In how many ways can you choose 1 per week for 6 weeks.
That's the same as the number of ways to select 6 items all at once, then read them once a week.
20P6
Unless you are allowed to choose the same book more than once. Then the number is 20^6
To determine the number of ways to choose 1 book per week for 6 weeks from a list of 20 books, we can use the concept of combinations.
The number of combinations can be calculated using the formula for combinations, which is given by:
C(n, k) = n! / (k! * (n - k)!)
where n is the total number of items and k is the number of items chosen.
In this case, we want to choose 1 book per week for 6 weeks, so n = 20 (total number of books) and k = 6.
Plugging in the values into the formula, we get:
C(20, 6) = 20! / (6! * (20 - 6)!)
Now, let's break down the calculation step by step:
1. Calculate the factorials:
- 20! = 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
- 6! = 6 * 5 * 4 * 3 * 2 * 1
- (20 - 6)! = 14! = 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
2. Simplify the expression:
- 20! / (6! * (20 - 6)!) = (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
3. Cancel out the common terms:
- (20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1))
= (20 * 19 * 18 * 17 * 16 * 15) / (6 * 5 * 4 * 3 * 2 * 1)
4. Evaluate the expression:
- (20 * 19 * 18 * 17 * 16 * 15) / (6 * 5 * 4 * 3 * 2 * 1) = 38,760
Therefore, there are 38,760 ways to choose 1 book per week for 6 weeks from a list of 20 books.