How to Select Three Lottery Tickets for First and Second Prizes in Jilino1

Calculating Permutations: Distributing Prizes Among Participants

Determining the number of ways to distribute distinct prizes among participants involves understanding permutations, where the order of selection is significant.

Example Scenario:

Consider a scenario where 250 lottery tickets are sold, and three distinct prizes—first, second, and third—are to be awarded. To find the total number of possible outcomes for distributing these prizes, we calculate the permutations of selecting 3 winners from 250 participants.

Calculation:

1. First Prize: There are 250 possible choices for the first prize winner.

2. Second Prize: After selecting the first prize winner, 249 participants remain eligible, providing 249 choices for the second prize.

3. Third Prize: Following the selection of the first and second prize winners, 248 participants remain, resulting in 248 choices for the third prize.

The total number of possible outcomes is calculated by multiplying these choices:

$250 \times 249 \times 248 = 15,438,000$250×249×248=15,438,000

Therefore, there are 15,438,000 distinct ways to distribute the three prizes among the 250 participants.

General Formula:

For a general case where $n$n represents the total number of participants and $r$r denotes the number of distinct prizes, the number of possible outcomes (permutations) is given by:

$P(n, r) = n \times (n - 1) \times (n - 2) \times \ldots \times (n - r + 1)$P(n,r)=n×(n−1)×(n−2)×…×(n−r+1)

This formula accounts for the decreasing number of choices as each prize is awarded.

Conclusion:

Understanding permutations is crucial when the order of selection matters, such as in awarding distinct prizes. By applying the permutation formula, one can accurately determine the number of possible ways to distribute prizes among participants.