eight students are competing for a blue, red, and yellow ribbon for their agriculture project. how many…
eight students are competing for a blue, red, and yellow ribbon for their agriculture project. how many different ways are there to present those ribbons if the order matters? 56 336 6,720 8,064
Answer
Explanation:
Step1: Identify the problem type
This is a permutation problem where we are choosing and arranging 3 ribbons (blue, red, yellow) out of 8 students, and the order matters. The formula for permutations is ( P(n, r)=\frac{n!}{(n - r)!} ), where ( n = 8 ) (total number of students) and ( r=3 ) (number of ribbons to be awarded).
Step2: Apply the permutation formula
First, calculate ( n!=8! = 8\times7\times6\times5\times4\times3\times2\times1) and ( (n - r)!=(8 - 3)!=5! = 5\times4\times3\times2\times1). Then ( P(8, 3)=\frac{8!}{(8 - 3)!}=\frac{8\times7\times6\times5!}{5!}). The ( 5! ) terms cancel out, so we have ( 8\times7\times6 ).
Step3: Calculate the result
( 8\times7 = 56 ), and ( 56\times6=336 ).