Question

Nine cars (3 Pontiacs [labeled 1-3], 4 Fords [labeled 4-7], and 2 Chevrolet's [labeled 8-9]) are divided into 3 groups for car racing. each group consistes of 3 cars, and they are allocated to 3 tracks (1-3), respectively.
1. How many different ways can you arrange the 9 cars?

Answer 1

Answer:

The number of different ways to arrange the 9 cars is 362,880.

Step-by-step explanation:

There are a total of 9 cars.

These 9 cars are to divided among 3 racing groups.

The condition applied is that there should be 3 cars in each group.

Use permutation to determine the total number of arrangements of the cars.

• For group 1:

       There are 9 cars and 3 to be allotted to group 1.

       This can happen in $^9P_{3}$ ways.

        That is, $^9P_{3}=\frac{9!}{(9-3)!} =504$ ways.

• For group 2:

       There are remaining 6 cars and 3 to be allotted to group 2.

       This can happen in $^6P_{3}$ ways.

        That is, $^6P_{3}=\frac{6!}{(6-3)!} =120$ ways.

• For group 3:

       There are remaining 3 cars and 3 to be allotted to group 3.

       This can happen in $^3P_{3}$ ways.

        That is, $^3P_{3}=\frac{3!}{(3-3)!} =6$ ways.

The total number of ways to arrange the 9 cars is: $^9P_{3}\times ^6P_{3}\times ^3P_{3}=504\times120\times6=362880$

Thus, the number of different ways to arrange the 9 cars is 362,880.