Question

Forty-three race cars started the 2004 Daytona 500 . How many ways can the cars finish first, second, and third? 10) There are 12 people entered in a swimming race. Assuming that there are no ties, in how many different ways can these people finish first, second, and third? 11) A landscaper wants to plant four oak trees, eight maple trees, and six poplar trees along the border of a lawn. The trees are to be spaced evenly apart. In how many distinguishable ways can they be planted?

Answer 1

There are 73,356 ways for the race cars to finish first, second, and third. There are 1,320 ways for the people to finish first, second, and third. There are 18! ways for the trees to be planted.

Step-by-step explanation:

For the first question, the number of ways the cars can finish first, second, and third can be calculated using the concept of permutations. Since there are 43 cars, the first place can be filled by any of the 43 cars. After the first car finishes, only 42 cars are left to choose from for the second place, and then only 41 cars are left for the third place. Therefore, the total number of ways the cars can finish is 43 × 42 × 41 = 73,356 ways.

For the second question, since there are 12 people, the first place can be filled by any of the 12 people. After the first person finishes, only 11 people are left to choose from for the second place, and then only 10 people are left for the third place. Therefore, the total number of ways the people can finish is 12 × 11 × 10 = 1,320 ways.

For the third question, the number of ways the trees can be planted can be calculated using the concept of combinations. Since there are 4 oak trees, 8 maple trees, and 6 poplar trees, the total number of trees is 4 + 8 + 6 = 18. Therefore, the number of ways the trees can be planted is 18! = 18 factorial.

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