A driver comes to pick up members of your family for a family reunion. The van holds 7 people, not including
the driver. Assuming that none of the babies will be riding in the van, how many different ways can 7 people be
chosen to ride in the van?
How many are in the family?
20
To solve this problem, we can use the concept of combinations. A combination is a selection of items where the order doesn't matter. In this case, we want to select 7 people to ride in the van out of a group of family members, assuming none of the babies will be riding.
The formula for combinations is:
C(n, r) = n! / (r!(n-r)!)
Where:
- n is the total number of items
- r is the number of items to be selected
- "!" denotes the factorial of a number, which means multiplying all positive integers from 1 to that number
In this problem, we have:
- n = total number of family members (excluding babies) = total number of adults
- r = number of people to be selected to ride in the van = 7 (since the van can hold 7 people)
Now, let's calculate the number of combinations:
C(n, r) = n! / (r!(n-r)!)
Since we know that there are no babies riding and the van holds 7 people, the total number of adults is 7. So, n = 7.
Plugging these values into the formula:
C(7, 7) = 7! / (7!(7-7)!)
Using the factorial formula:
7! = 7 x 6 x 5 x 4 x 3 x 2 x 1
Simplifying the expression:
C(7, 7) = (7 x 6 x 5 x 4 x 3 x 2 x 1) / [(7 x 6 x 5 x 4 x 3 x 2 x 1) x (7-7)!]
C(7, 7) = 5040 / (5040 x 0!)
Since any number divided by zero is undefined, we can see that (7-7)! is equal to 0!.
So, the result is:
C(7, 7) = 5040 / (5040 x 0!)
Therefore, there is only one way to select 7 people to ride in the van, assuming none of the babies will be riding.