here are 13 female board members and 17 male board members.
How many ways are there to make a committee of 18 board members?
How many ways are there to make a committee of 18 board members if exactly 6 must be female?
To calculate the number of ways to make a committee of 18 board members, we will use the combination formula.
1. First, let's find the number of ways to select 18 board members without any restrictions on gender:
nCr = n! / [(n-r)! * r!]
In this case, n = 30 (13 female board members + 17 male board members) and r = 18.
The equation becomes:
30C18 = 30! / [(30-18)! * 18!]
Solving this will give us the number of ways to make a committee of 18 board members.
2. Now let's find the number of ways to make a committee of 18 board members where exactly 6 must be female:
We need to break down this problem into two parts:
a) Selecting the 6 female board members:
nCr = n! / [(n-r)! * r!]
n = 13 (number of female board members) and r = 6.
The equation becomes:
13C6 = 13! / [(13-6)! * 6!]
b) Selecting the remaining 12 board members from the remaining pool (30 - 6) = 24 board members (17 male + 7 remaining female board members):
nCr = n! / [(n-r)! * r!]
n = 24 (number of remaining board members) and r = 12.
The equation becomes:
24C12 = 24! / [(24-12)! * 12!]
Now, to find the total number of ways, we multiply the results from a) and b). This gives us the number of ways to make a committee of 18 board members with exactly 6 females.
Please let me know if you need any further step-by-step explanation.
To find the number of ways to make a committee of 18 board members, we can use the concept of combinations.
For the first question, where there are no restrictions on the composition of the committee, we can select 18 members out of a total of 30 members (13 females + 17 males).
The formula for combinations is given by:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of members (females + males), and r is the number of members we want to select (18 in this case).
Using this formula, we can calculate the number of ways to make a committee of 18 board members:
C(30, 18) = 30! / (18! * (30-18)!)
= 30! / (18! * 12!)
= 30 * 29 * 28 * ... * 13 / 18 * 17 * 16 * ... * 1
Now, let's move on to the second question, where exactly 6 members must be female.
We need to select 6 females out of 13 and 12 males out of 17, in order to have a total of 18 members.
Using the same combination formula, we can calculate the number of ways to select these members:
C(13, 6) * C(17, 12)
= (13! / (6! * (13-6)!) * (17! / (12! * (17-12)!))
= (13 * 12 * 11 * ... * 8) / (6 * 5 * 4 * ... * 1) * (17 * 16 * 15 * ... * 6) / (12 * 11 * 10 * ... * 1)
Therefore, the number of ways to make a committee of 18 board members if exactly 6 must be female is the product of these two combinations.
I hope this explanation helps you understand how to approach these types of problems.
again?
30C18
13C6 * 17C12