Question

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

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 \( 56\times6=336 \).

Answer:

336