Alfred, brandon and charles are the three participants in a race. in how many different ways can the three finish if it's possible for two or more participants to finish in a tie

There are 13 different ways to finish the race

Step-by-step explanation:

To identify the number of different ways to finish the race we should categorized the types of win:

1. Without tie: the ways that are possible in this option are defined by the rule of multiplicación. There are 3 participants and 3 places, there are 3 possibles participant that can occupy the first place, then 2 possibles participants that can occupy the second place and one participant that can occupy the last place.

       3               x          2                x          1                  =   6

first place              second place           third place

So, there are 6 different ways to win without tie.

2. Double tie in the second place: This time there are 3 participant that can occupy the first place and the other participant are going to occupy the second place. So there are only 3 ways in this option, every option is defined by the winner. that's mean: the winner is Alfred, or is Brandon or  is Charles.

3. Double tie in the first place: This option is like option 2, there are only 3 ways because every option is defined by the person that is in the second place. That's mean: the second place is occupy by Alfred, or by Brandon or by Charles.

4. All the participants end in tie: There are just one possibility in this option, all are the winners.

Finally there are 13 ways to end this race, 6 ways without tie, 3 ways with double tie in second place, 3 ways with double tie in first place and 1 way with all the participants in the same place.

256π units²   (answer b)

step-by-step explanation:

the formula for the surface area of a sphere of radius r

is             a = 4πr²

and so, if r = 8 units, then the surface area is:

a = 4π(8 units)² = 256π units²   (answer b)