2. If you have *CHOOSE A NUMBER BETWEEN 7 and 20* people, how many ways are there to:
I chose 9
a. Arrange them in a line
362880 ways to line them up
b. Arrange them in a circle
40320 ways
c. Select a president, vice president and secretary?
I think the answer is 504
To find the number of ways to arrange a group of people, whether in a line or a circle, we can use the concept of permutations. A permutation is an arrangement of objects in a specific order.
a. Arranging them in a line:
To calculate the number of ways to arrange the people in a line, we need to use the formula for permutations. The formula for permutations is n!, where n represents the number of objects.
In this case, you chose 9 people, so n = 9. Therefore, the number of ways to arrange them in a line is 9!.
To calculate 9!, multiply all the integers from 1 to 9 together:
9! = 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880 ways.
So, there are 362,880 ways to arrange the 9 people in a line.
b. Arranging them in a circle:
When arranging objects in a circle, we need to consider that any rotation of the arrangement is considered the same arrangement. So, for a group of n objects arranged in a circle, the number of ways to arrange them is (n-1)!.
In this case, you still have 9 people, so n = 9. Therefore, the number of ways to arrange them in a circle is (9-1)!
To calculate (9-1)!, multiply all the integers from 1 to 8 together:
8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320 ways.
So, there are 40,320 ways to arrange the 9 people in a circle.
c. Selecting a president, vice president, and secretary:
To calculate the number of ways to select individuals for different positions, we use the concept of combinations. A combination is the selection of objects without considering their order.
In this case, you still have 9 people to choose from, but you need to select a president, vice president, and secretary. To calculate the number of ways to select them, we can use the formula for combinations, which is nCr, where n represents the total number of objects available, and r represents the number of objects to be selected.
In this case, you need to select 3 people from a group of 9, so n = 9 and r = 3. Therefore, the number of ways to select a president, vice president, and secretary is 9C3.
To calculate 9C3, you can use the formula:
9C3 = 9! / (3! * (9-3)!)
Plug in the values and compute:
9C3 = 9! / (3! * 6!) = 9 x 8 x 7 / (3 x 2 x 1) = 84
So, there are 84 ways to select a president, vice president, and secretary from a group of 9 people.