In a batch of 97 computer diskettes, 8 are defective. A sample of six diskettes is to be selected from the batch.How many of the samples consist of all defective diskettes?
To determine how many of the samples consist of all defective diskettes, we can use the concept of combinations.
First, let's calculate the number of ways we can choose 6 defective diskettes from the 8 defective diskettes. This can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
where:
C(n, r) represents the number of combinations.
n represents the total number of items.
r represents the number of items we want to choose.
In this case, we have 8 defective diskettes and we want to choose all of them (r = 8). So, the formula becomes:
C(8, 8) = 8! / (8!(8-8)!)
Since 8! / 8! equals 1 and 8 - 8 equals 0, the expression simplifies to:
C(8, 8) = 1
Therefore, there is only one way to choose all the defective diskettes from the 8 available.
Next, we need to determine how many samples of size 6 we can choose from the batch of 97 diskettes. For this, we'll use the same combination formula, except now we have a total of 97 diskettes and we want to choose 6:
C(97, 6) = 97! / (6!(97-6)!)
To calculate this value, we evaluate the expression:
C(97, 6) = 97! / (6!(91)!)
It is worth mentioning that 91 equals 97 - 6.
After simplifying the expression, we find:
C(97, 6) = 15,409,840
Therefore, there are 15,409,840 different samples of 6 diskettes that can be chosen from the batch of 97.
To answer the question, we know there is only one way to choose all defective diskettes and 15,409,840 different samples of 6 diskettes in total. Hence, there is only one sample that consists of all defective diskettes.