In how many ways can a 5 member committee be formed from a group of 40?
a) if there are only 8 that can be a leader, how many different groupings with a leader can be formed?
first part is simply 40 choose 5
= C(40,5) = 658008
b) The leader can be picked in 8 ways, and then we have to choose 4 more from 39
= 8 C(39,4) = 658008
Are you surprised that the answers are the same ?
A group of 4 doctors, 9 teachers, and 11 business leaders are going to form a committee of two. How many different ways can the committee be formed if it doesn't matter what the person's profession is?
To find the number of ways to form a committee from a group, we can use the concept of combinations.
1) For the first part of the question, where we need to find the number of ways to form a 5-member committee from a group of 40 without any specifications:
We can use the formula for combinations, also known as "nCr" (n choose r).
The formula for combinations is given by:
nCr = n! / (r! * (n-r)!)
Here, n is the total number of members in the group (40), and r is the number of members we want to choose for the committee (5).
Using this formula, we can calculate the number of ways to form the committee without any restrictions:
40C5 = 40! / (5! * (40-5)!)
Now, let's calculate the value using the factorial function on a calculator or programming language:
40! = 40 × 39 × 38 × 37 × 36 × 35 × 34 × 33 × 32 × 31 × 30 × 29 × 28 × 27 × 26 × 25 × 24 × 23 × 22 × 21 × 20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
5! = 5 × 4 × 3 × 2 × 1
35! = 35 × 34 × 33 × 32 × 31 × 30 × 29 × 28 × 27 × 26 × 25 × 24 × 23 × 22 × 21 × 20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
Substituting the values into the formula:
40C5 = 40! / (5! * 35!)
Now, you can simplify and calculate the value to get the answer.
2) For the second part, where we need to find the number of different groupings with a leader:
Since there are 8 members who can be the leader, we can choose one of the 8 leaders and then form the remaining committee of 4 members from the remaining 39 members.
Using the same formula for combinations:
39C4 = 39! / (4! * (39-4)!)
Substituting the values into the formula and calculating the value will give you the answer.
I hope this explanation helps in understanding how to solve the problem!