Question

A researcher conducts a random selection of 3 fish from a total of 10 fish in a tank, placing each of the selected fish into separate containers. Determine the number of distinct ways in which this selection and placement can be accomplished. Provide a detailed explanation of the counting process involved in arriving at the final result.

Answer 1

There are 720 distinct ways to select and place 3 fish out of 10 into separate containers because the order of placement matters, which is calculated using the permutation formula P(10, 3) = 10! / (10-3)!.

Step-by-step explanation:

The question involves the concept of permutations because the order of selection and placement matters. To determine the number of distinct ways in which a researcher can select and place 3 fish from a total of 10 into separate containers, we use the permutation formula which is, in this case, P(10, 3) where P(n, r) = n! / (n-r)!. For 10 fish, we calculate the permutation as follows:

1. Calculate the factorial of the total number of fish, which is 10!.
2. Determine the factorial of the number of fish not selected, which is (10-3)! or 7!.
3. Divide the factorial of 10 by the factorial of 7 to get the total number of unique permutations: 10! / 7! = (10 x 9 x 8 x 7!) / 7! = 720 distinct ways.