How many different ways are there to arrange 10 people in a row of 10 chairs?
The correct answer and explanation is:
The number of different ways to arrange 10 people in a row of 10 chairs is given by the total number of permutations of the 10 individuals. This is a simple case of finding the factorial of 10, written as 10!10!.
The factorial of a number is the product of all positive integers from 1 to that number. In this case, the factorial of 10 is: 10!=10×9×8×7×6×5×4×3×2×1=3,628,80010! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
Therefore, there are 3,628,800 different ways to arrange 10 people in a row of 10 chairs.
Explanation:
To understand this, let’s break it down step by step:
- First Chair: There are 10 people available, so for the first chair, there are 10 possible choices of who can sit in it.
- Second Chair: After the first person is seated, only 9 people remain. Thus, there are 9 choices for who can sit in the second chair.
- Third Chair: Now, with 2 people already seated, only 8 people remain. Therefore, there are 8 choices for the third chair.
This pattern continues until all the chairs are filled. The total number of ways to arrange the 10 people is the product of these choices: 10×9×8×7×6×5×4×3×2×1=3,628,80010 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
This product is also known as 10 factorial (10!). Factorials are used in permutations and combinations to count possible arrangements when the order matters.
Thus, the number of different ways to arrange the 10 people in 10 chairs is 3,628,800.