Question

1. a researcher randomly selects 4 fish from among 8 fish in a tank and puts each of the 4 selected fish into different containers. how many ways can this be done? a) 128 b) 420 c) 1680 d) 6720 e) 8400 f) none of the above

Explanation:

Step1: Identify the problem type

This is a permutation problem where we select 4 fish from 8 and arrange them into 4 different containers (order matters as containers are different). The formula for permutations is \( P(n, r)=\frac{n!}{(n - r)!} \), where \( n = 8 \) (total fish) and \( r = 4 \) (fish to select and arrange).

Step2: Calculate the permutation

Substitute \( n = 8 \) and \( r = 4 \) into the formula:
\( P(8, 4)=\frac{8!}{(8 - 4)!}=\frac{8!}{4!} \)
Since \( 8! = 8\times7\times6\times5\times4! \), we can cancel \( 4! \) from numerator and denominator:
\( P(8, 4)=8\times7\times6\times5 = 1680 \)

Answer:

C) 1680