Suppose that 10 dice are rolled. Each die is a regular 6 -sided die with numbers 1 through 6 labelled on the sides. How many different sums of all 10 numbers are possible?
smallest sum is 10, largest sum of 600
since all possible sums between are possible,
all we have to do is find which term number 600 is
of the arithmetic sequence with
term1 = 10 , d = 1, term(n) = 600
a + (n-1)d = term(n)
10 + n-1 = 600
n-1 = 590
n = 591
There are 591 different sums possible.
To find the number of different sums possible when rolling the 10 dice, we need to consider the range of possible outcomes for each die.
A single die has 6 possible outcomes, ranging from 1 to 6.
When we roll the first die, we can get a sum of any of these 6 numbers. Now, when we roll the second die, for each outcome of the first die, we can get 6 additional possible sums (since each outcome of the second die can be combined with each outcome of the first die).
So, by the time we roll the second die, there are already 6 different sums possible.
For each additional die, the number of possible sums doubles, because for each previous sum, we have 6 new possibilities. Therefore, after rolling the third die, we'll have 6 x 2 = 12 different sums.
We can continue this pattern until we roll all 10 dice:
1 die: 6 sums
2 dice: 6 x 2 = 12 sums
3 dice: 6 x 2 x 2 = 24 sums
4 dice: 6 x 2 x 2 x 2 = 48 sums
....
10 dice: 6 x 2^9 = 6 x 512 = 3072 sums
Therefore, when rolling 10 dice, there are 3072 different possible sums.