Suppose that we wish to seat ten people, 5 men and 5 women, in a row of ten chairs. In how many ways can we seat the people with no restrictions?
mmmh, with no restriction?
that would be just 10!
Thanks. I got it. The answer is 3,628,800
To determine the number of ways we can seat the 10 people with no restrictions, we can use the concept of permutations.
In this scenario, we have 10 individuals (5 men and 5 women) to seat in 10 chairs.
Step 1: Identify the number of options for the first chair.
Since we have 10 people to choose from, there are 10 options for the first chair.
Step 2: Determine the number of options for the second chair.
After placing someone in the first chair, we have 9 people remaining to choose from for the second chair.
Step 3: Continue this process for each remaining chair.
Keep reducing the number of people available by 1 for each subsequent chair until all 10 chairs are filled.
Step 4: Find the total number of ways.
To find the total number of ways, we multiply the number of options at each step:
10 options for the first chair * 9 options for the second chair * 8 options for the third chair * ... * 2 options for the ninth chair * 1 option for the tenth chair.
Mathematically, this can be represented as:
10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1
Calculating this value gives us: 3,628,800
Therefore, there are 3,628,800 ways to seat 10 people (5 men and 5 women) with no restrictions.
To find the number of ways to seat ten people, 5 men and 5 women, in a row of ten chairs with no restrictions, we can use the concept of permutation.
Permutation is used when the order of arrangement matters. In this scenario, the order of seating is important because each person will occupy a different chair.
Since there are 10 chairs available, we can start by placing the first person in any of the 10 chairs. After that, there will be 9 chairs left for the second person, 8 chairs left for the third person, and so on.
To find the total number of ways, we can multiply the number of choices for each person.
For the first person, there are 10 choices. For the second person, there are 9 choices left. Continuing this pattern, for the third person, there are 8 choices, and so on.
Therefore, the total number of ways to seat all ten people with no restrictions is:
10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 3,628,800
Hence, there are 3,628,800 ways to seat the ten people in a row of ten chairs with no restrictions.