+19 Three standard -sided dice are colored magenta, yellow, and cyan. In how many ways can these dice be rolled so the numbers on the faces sum to a total of ? (The dice are distinguishable, so a magenta yellow and cyan is different from a magenta yellow and cyan )
We can solve this problem using complementary counting. We can count the number of ways to roll the dice so that the sum is not 20, and then subtract from the total number of ways to roll the dice.
There are 610 ways to roll the dice in general. If the sum is not 20, then there must be at least one die that rolls a 1. There are 510 ways to roll the dice so that exactly one die rolls a 1, and 410 ways to roll the dice so that exactly two dice roll 1s, and so on.
Note that if exactly k dice roll 1s, then the remaining 10−k dice must all roll 2s or higher. This is because the sum of the numbers on the faces must be at least 20, and if any of the dice rolled a 1, the sum would be less than 20.
Therefore, the number of ways to roll the dice so that the sum is not 20 is equal to: $∑k=110510−k⋅4k=510(1+4+42+⋯+410)$We can use the formula for a geometric series to evaluate this sum: $1+4+42+⋯+410=4−1411−1=63,488$Therefore, there are 610−510⋅63,488=492 ways to roll the dice so the numbers on the faces sum to a total of 20.
