In the Cash Now lottery game there are 10 finalists who...

In the Cash Now lottery game there are 10 finalists who submitted entry tickets on time. From these 10 tickets, three grand prize winners will be drawn. The first prize is $\$ 1 dollars million, the second prize is $\$ 100,000 dollars and the third prize is $\$ 10,000 .$ Determine the total number of different ways in which the winners can be drawn. (Assume that the tickets are not replaced after they are drawn.)

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Key Concepts

Permutations

A permutation is an arrangement of objects in a specific order, and it is a crucial concept when the order of selection matters, such as assigning distinct prizes to a set of finalists. Since each prize is unique, the sequence in which winners are selected is important, making the problem one of counting permutations rather than combinations.

Multiplication Principle

The multiplication principle (or counting principle) is used to calculate the total number of outcomes when there are several sequential decisions to be made. In the context of selecting winners one after the other without replacement, the number of choices available decreases with each selection, and the total number of arrangements is found by multiplying the number of choices at each step.

Counting without Replacement

Counting without replacement refers to the process of drawing items from a set such that once an item is selected, it is not put back, thereby reducing the total count for subsequent selections. This is essential in lottery scenarios where each selected ticket cannot win another prize, affecting the total number of possible distinct winner arrangements.

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