Suppose you are managing 16 employees, and you need to form three teams to work on different projects. Assume that all employees will work on a team, and that each employee has the same qualifications/skills so that everyone has the same probability of getting choosen. In how many different ways can the teams be chosen so that the number of employees on each project are as follows:
7, 1, 8
16! / (7! * 1! * 8!)
To determine the number of ways to form three teams with 7, 1, and 8 employees, we can use the concept of combinations.
Step 1: Selecting the team with 7 employees:
We need to choose 7 employees from a group of 16. The number of ways to do this is given by the combination formula C(16, 7).
C(16, 7) = 16! / (7! * (16-7)!)
= 16! / (7! * 9!)
= (16 * 15 * 14 * 13 * 12 * 11 * 10) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
= 11440
So, there are 11,440 ways to select the team with 7 employees.
Step 2: Selecting the team with 1 employee:
We have 8 remaining employees to choose from, and we only need to select 1 employee for this team. Therefore, the number of ways to do this is C(8, 1) = 8.
Step 3: Selecting the team with 8 employees:
Since all the remaining employees (8) are assigned to this team, there is only one way to select them.
Finally, to find the total number of ways to form the teams, we multiply the results from each step together:
Total number of ways = 11,440 * 8 * 1
= 91,520
Therefore, there are 91,520 different ways to form three teams with 7, 1, and 8 employees.
To find the number of different ways to form the teams, we need to use the concept of combinations.
First, let's choose the 7 employees for the first team. We can do this by calculating the combination of 16 employees taken 7 at a time. This can be represented as C(16, 7).
The formula for combinations is given by: C(n, r) = n! / (r! * (n-r)!), where n represents the set from which you are choosing and r represents the number of elements you are choosing.
So, to find C(16, 7), we perform the following calculations:
16! / (7! * (16-7)!)
= 16! / (7! * 9!)
= (16 * 15 * 14 * 13 * 12 * 11 * 10) / (7 * 6 * 5 * 4 * 3 * 2 * 1)
= 16 * 15 * 14 * 13 * 12 * 11 * 10 / 7 * 6 * 5 * 4 * 3 * 2 * 1
= 11440
So, there are 11,440 different ways to choose 7 employees from a pool of 16.
Now let's move on to the second team. We need to select only 1 employee. Since we have already chosen 7 employees for the first team, the pool of employees from which we can choose for the second team is reduced to 16 - 7 = 9 employees.
So, the number of ways to choose 1 employee from 9 can be calculated as C(9, 1) = 9! / (1! * (9-1)!) = 9.
Finally, for the third team, we need to choose 8 employees. We have already chosen 7 employees for the first team and 1 employee for the second team, leaving us with 16 - 7 - 1 = 8 employees in the pool.
So, the number of ways to choose 8 employees from 8 can be calculated as C(8, 8) = 8! / (8! * (8-8)!) = 1.
Now, to find the total number of ways to form the teams with the given distribution of employees, we multiply the number of ways for each team:
Total number of ways = number of ways for the first team * number of ways for the second team * number of ways for the third team
= 11440 * 9 * 1
= 102,960
Therefore, there are 102,960 different ways to form three teams with 7, 1, and 8 employees each respectively, given a pool of 16 employees.
16C8 * 8C7
Well, there are certainly many ways to form these teams, but let's see if we can clown around with the math a little bit.
So, to form the first team of 7 employees, we need to choose 7 out of the 16 employees. This can be calculated using combinations, which is represented by the symbol "C". So the number of ways to choose 7 employees out of 16 is written as 16C7. Using some fancy math, we can calculate this as:
16C7 = 16! / (7! * (16-7)!) = (16*15*14*13*12*11*10) / (7*6*5*4*3*2*1) = 11440
Now, we need to select 1 employee for the second team. Again, we only have one spot to fill, so we need to choose 1 out of the remaining 9 employees (16 - 7 = 9). Similarly, we can calculate this as 9C1 = 9.
Finally, for the third team of 8 employees, we need to choose 8 out of the remaining 8 employees (16 - 7 - 1 = 8). Again, using combinations, we get 8C8 = 1.
Now, to find the total number of different ways to form the teams, we just multiply the number of possibilities for each team:
11440 * 9 * 1 = 103,040
So, there are 103,040 different ways to form three teams with the specified number of employees on each project. And that's no joke!