In Major League Baseball, there are five teams in the Eastern Division of the National League: Atlanta, Florida, New York, Philadelphia, and Washington. How many different orders of finish are there for these five teams? (Assume that there are no ties in the standings.) Do you believe that all these orders are equally likely? Discuss.

There are 120 different orders of finish for these five teams.

These orders are not equally as likely. For example, let's look back at the 2019 season. Washington won the World Series, Atlanta made the playoffs, New York and Philadelphia kind of treaded water and Florida(now Miami) is rebuilding and were very bad. So the talent level of the teams change the probabilities. For example, Washington and Atlanta have higher probabilities to finish first than the other 3 teams, while it is also quite likely that Philadelphia and New York finish ahead of Miami.

Step-by-step explanation:

How many different orders of finish are there for these five teams?

There are 5 teams.

One of them will finish in first place. So there are 5 possible teams for first place.

One of them will finish in second place. All of them can finish in 2nd place, except the one which finished first. So there are 4 possible teams for second place.

One of them will finish in third place. Those who finished in 1st and 2nd cannot finish in 3rd. So there are 3 possible teams for third place.

Following this logic, there are two possible teams for fourth place and one for fifth.

5*4*3*2*1 = 120

There are 120 different orders of finish for these five teams.

Do you believe that all these orders are equally likely? Discuss.

In practice no.

For example, let's look back at the 2019 season. Washington won the World Series, Atlanta made the playoffs, New York and Philadelphia kind of treaded water and Florida(they are now called Miami Marlins) is rebuilding and were very bad. So the talent level of the teams change the probabilities. For example, Washington and Atlanta have higher probabilities to finish first than the other 3 teams, while it is also quite likely that Philadelphia and New York finish ahead of Miami.

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."