a) A baking club wants to form an executive committee. There are 12 people in the baking club, including Mark. In how many ways can the baking club form an executive committee with 2 people?
b) A baking club wants to form an executive committee. There are 12 people in the baking club, including Mark. In how many ways can the baking club form an executive committee with 2 people, including Mark?
c) A baking club wants to form an executive committee. There are 12 people in the baking club, including Mark. In how many ways can the baking club form an executive committee with 2 people, not including Mark?
a) To find the number of ways the baking club can form an executive committee with 2 people, we need to choose 2 people out of the 12 people. This can be done in mathematical notation as 12C2.
Using the formula for combinations, 12C2 = 12! / (2! * (12-2)!)
Simplifying this expression, we get: 12! / (2! * 10!)
Since 2! = 2, this further simplifies to: 12 * 11 / 2
Therefore, there are 66 ways the baking club can form an executive committee with 2 people.
b) Since Mark must be included in the executive committee, we only need to choose 1 more person from the remaining 11 people. This can be done in mathematical notation as 11C1.
Using the formula for combinations, 11C1 = 11! / (1! * (11-1)!)
Simplifying this expression, we get: 11! / (1! * 10!)
Since 1! = 1, this further simplifies to: 11 * 1
Therefore, there are 11 ways the baking club can form an executive committee with 2 people, including Mark.
c) To form an executive committee with 2 people, not including Mark, we need to choose 2 people from the remaining 11 people (excluding Mark). This can be done in mathematical notation as 11C2.
Using the formula for combinations, 11C2 = 11! / (2! * (11-2)!)
Simplifying this expression, we get: 11! / (2! * 9!)
Since 2! = 2, this further simplifies to: 11 * 10 / 2
Therefore, there are 55 ways the baking club can form an executive committee with 2 people, not including Mark.
a) To form an executive committee with 2 people, we need to choose 2 people out of the 12 people in the baking club. This can be done using the combination formula.
The number of ways to choose 2 people out of 12 is given by:
C(12, 2) = 12! / (2! * (12-2)!)
Simplifying, we get:
C(12, 2) = 12! / (2! * 10!)
C(12, 2) = (12 * 11) / (2 * 1)
C(12, 2) = 66
So, there are 66 ways the baking club can form an executive committee with 2 people.
b) Since Mark is included in the committee, we need to choose only 1 more person from the remaining 11 people in the baking club.
The number of ways to choose 1 person out of 11 is given by:
C(11, 1) = 11! / (1! * (11-1)!)
Simplifying, we get:
C(11, 1) = 11! / (1! * 10!)
C(11, 1) = 11
So, there are 11 ways the baking club can form an executive committee with 2 people, including Mark.
c) Mark is not included in the committee, so we need to choose 2 people from the remaining 11 people in the baking club.
The number of ways to choose 2 people out of 11 is given by:
C(11, 2) = 11! / (2! * (11-2)!)
Simplifying, we get:
C(11, 2) = 11! / (2! * 9!)
C(11, 2) = (11 * 10) / (2 * 1)
C(11, 2) = 55
So, there are 55 ways the baking club can form an executive committee with 2 people, not including Mark.
To answer these questions, we can use a combination formula, specifically the formula for "n choose r." The formula for "n choose r" is:
C(n, r) = n! / (r!(n-r)!)
Where n is the total number of items or people to choose from, and r is the number of items or people to choose.
Now let's apply this formula to each part of the question:
a) To form an executive committee with 2 people, we need to choose 2 people from a total of 12 people. Using the combination formula:
C(12, 2) = 12! / (2!(12-2)!)
= 12! / (2!10!)
Simplifying further:
C(12, 2) = (12 * 11 * 10!) / (2!10!)
= (12 * 11) / 2!
= (12 * 11) / 2
= 66
Therefore, there are 66 ways the baking club can form an executive committee with 2 people.
b) In this case, we want to include Mark in the executive committee. So we have to choose 1 person from the remaining 11 people (excluding Mark) to form the committee. Using the combination formula:
C(11, 1) = 11! / (1!(11-1)!)
= 11! / (1!10!)
Simplifying further:
C(11, 1) = (11 * 10!) / (1!10!)
= (11 * 10) / 1
= 110
Therefore, there are 110 ways the baking club can form an executive committee with 2 people, including Mark.
c) Here, we need to choose 2 people from the 11 people excluding Mark. Using the combination formula:
C(11, 2) = 11! / (2!(11-2)!)
= 11! / (2!9!)
Simplifying further:
C(11, 2) = (11 * 10 * 9!) / (2!(9!))
= (11 * 10) / 2
= 55
Therefore, there are 55 ways the baking club can form an executive committee with 2 people, not including Mark.