hi, could someone please help me solve this problem, i know the answer is 24, i just need to know how to solve it.
a "lucky dollar" is one of nine symbols on each reel of a slot machine with three reels. a player receives one of the various payouts whenever one or more lucky dollars appear in the window of the machine. find the number of winning combinations for which the machine has a payoff.
$ in first slot with 81 possibilities for the next two
$ in the second slot with 8 possibilities in the left slot and 9 in the right slot = 72 (Double counting must be avoided; hence the 8)
$ in the third slot with 8 possibilities in the left slot and 8 in the middle slot = 64.
Total = 81 + 72 + 64 = 217 possible payout combinations (out of 9^3 = 729)
To find the number of winning combinations for which the machine has a payoff, we need to determine the different ways the lucky dollar symbol can appear on the reels.
Since there are 9 symbols on each reel, there are 9 possible outcomes for each reel. As there are 3 reels, the total number of combinations is calculated by multiplying the possibilities for each reel: 9 * 9 * 9 = 729.
However, we need to subtract the possibility of all blanks appearing on the reels, as this would not result in a payoff. There is only 1 blank symbol option on each reel, so the number of combinations where all blanks appear is 1 * 1 * 1 = 1.
Therefore, the number of winning combinations for which the machine has a payoff is 729 - 1 = 728.
Hence, the answer is 728.
To solve this problem, we need to calculate the number of winning combinations for which the machine has a payoff.
Given:
- Each reel of the slot machine has nine symbols, including the "lucky dollar."
- The machine has three reels.
To find the number of winning combinations, we need to consider the different scenarios where one or more lucky dollars appear in the window of the machine.
Scenario 1: One lucky dollar appears in the window.
To calculate the number of winning combinations for this scenario, we need to consider all the possible positions where the lucky dollar can appear on each of the three reels.
- For the first reel, there are 9 possible symbols, out of which one is the lucky dollar. So there is one way to have a lucky dollar in the first reel.
- Similarly, for the second reel, there is again one way to have a lucky dollar.
- And for the third reel, there is one way to have a lucky dollar.
Therefore, the total number of winning combinations for this scenario is 1 * 1 * 1 = 1.
Scenario 2: Two lucky dollars appear in the window.
In this scenario, we need to consider the different ways in which two lucky dollars can appear on the three reels.
- For the first and second reels, there are 1 * 1 = 1 way to have two lucky dollars.
- For the first and third reels, there are also 1 * 1 = 1 way to have two lucky dollars.
- And for the second and third reels, again there is 1 * 1 = 1 way to have two lucky dollars.
Therefore, the total number of winning combinations for this scenario is 1 + 1 + 1 = 3.
Scenario 3: Three lucky dollars appear in the window.
In this scenario, we have to consider the only way in which all three reels can have the lucky dollar, which is 1 * 1 * 1 = 1.
Therefore, the total number of winning combinations for this scenario is 1.
Now, to find the overall total, we need to sum up the winning combinations from all three scenarios:
Total = Scenario 1 + Scenario 2 + Scenario 3 = 1 + 3 + 1 = 5.
So, the number of winning combinations for which the machine has a payoff is 5.
By following this method, we can solve the given problem step by step and calculate the desired answer.