A sample of 5 people is selected from 3 smokers and 12 non-smokers. In how many ways can the 5 people be selected?
Since no restriction in terms of being a smoker or non-smoker is required, we are simply choosing 5 people from 15
which is C(15,5) = 3003
To find the number of ways the 5 people can be selected, we can use the concept of combinations.
In this case, we have 3 smokers and 12 non-smokers, and we want to select 5 people out of this group. We need to consider two different scenarios: selecting all smokers or selecting a mix of smokers and non-smokers.
Scenario 1: Selecting all smokers
Since there are only 3 smokers, we can select all of them in one way.
Scenario 2: Selecting a mix of smokers and non-smokers
Since we want to select 5 people in total, we need to choose some smokers and some non-smokers. To do this, we can consider the combinations of smokers and non-smokers.
The total number of ways to select 5 people from 15 (3 smokers + 12 non-smokers) can be found using the combination formula:
C(n, r) = n! / (r! * (n-r)!)
Where n is the total number of elements and r is the number of elements we want to choose.
In this case, n = 15 and r = 5. Plugging these values into the formula:
C(15, 5) = 15! / (5! * (15-5)!)
= 15! / (5! * 10!)
Calculating the factorial values:
15! = 15 * 14 * 13 * 12 * 11 * 10!
5! = 5 * 4 * 3 * 2 * 1
Substituting the factorial values into the formula:
C(15, 5) = (15 * 14 * 13 * 12 * 11 * 10!) / ((5 * 4 * 3 * 2 * 1) * 10!)
= (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
= (15 * 14 * 13 * 12 * 11) / (5!)
Simplifying the equation:
C(15, 5) = (15 * 14 * 13 * 12 * 11) / (5 * 4 * 3 * 2 * 1)
= 3003
Therefore, there are 3003 ways to select the 5 people from the given group of 3 smokers and 12 non-smokers.