Assume that the student
has a cup with 15 writing implements: 8 pencils, 5 ball
point pens, and 2 felt-tip pens.In how many ways can the selection be made if no more than one ball
point pen is selected?
To find the number of ways we can make the selection, we need to consider two scenarios: selecting a ball point pen or not selecting a ball point pen.
Scenario 1: Selecting a ball point pen
In this scenario, we can select 0 or 1 ball point pen. We have 5 ball point pens, so there are 2 options:
- Option 1: Select 0 ball point pens. In this case, we need to select a combination of pencils and felt-tip pens only. We have 8 pencils and 2 felt-tip pens, so we can select them in (8+2)C(15) = 10C15 ways using the combination formula.
- Option 2: Select 1 ball point pen. In this case, we have one less ball point pen and we need to select a combination of pencils, felt-tip pens, and the remaining ball point pens. We have 8 pencils, 2 felt-tip pens, and 4 remaining ball point pens. So we can select them in (8+2+4)C(15) = 14C15 ways.
Scenario 2: Not selecting a ball point pen
In this scenario, we can select a combination of pencils and felt-tip pens only. We have 8 pencils and 2 felt-tip pens, so we can select them in (8+2)C(15) = 10C15 ways.
To find the total number of ways, we sum up the number of ways from both scenarios:
Total number of ways = (Number of ways in Scenario 1 - Option 1) + (Number of ways in Scenario 1 - Option 2) + (Number of ways in Scenario 2)
Total number of ways = (10C15) + (14C15) + (10C15)
To calculate these combinations, we can use the formula for combination:
nCr = n! / (r!(n-r)!)
Plugging in the values and calculating each combination, we get:
Total number of ways = (10! / (15!(10-15)!)) + (14! / (15!(14-15)!)) + (10! / (15!(10-15)!))
Total number of ways = (1 / (15!(-5)!)) + (14 / (15!(14)!)) + (1 / (15!(-5)!))
Total number of ways = (1/3003) + (14/15) + (1/3003)
Total number of ways = 0.000333 + 0.9333 + 0.000333
Total number of ways = 0.933966
Therefore, there are approximately 0.933966 ways to make the selection if no more than one ball point pen is selected.