Scott repeatedly rolls a pair of standard six-sided dice and keeps track of the
sum of the two numbers rolled each time. So far he has not rolled any sum
twice. What is the maximum possible number of times he has thrown the dice?
**** 1 2 3 4 5 6
===================
1***2 3 4 5 6 7
2***3 4 5 6 7 8
3***4 5 6 7 8 9
4***5 6 7 8 9 10
5***6 7 8 9 10 11
6 **7 8 9 10 11 12
There are 11 sums from 2 to 12 so you can roll 11 times without repeating
Well, it's clear that Scott must be having a great time rolling those dice! But let's crunch some numbers to figure out the maximum possible number of times he has thrown them.
To find the maximum number of times Scott has rolled the dice, we need to consider the different sums we can get from rolling a pair of standard six-sided dice.
There are 11 possible sums, ranging from 2 to 12. Now, if Scott hasn't rolled any sum twice, that means he must have rolled 11 different sums.
Let's break down the numbers a bit more:
- If Scott rolls a 2, there's only one combination that gives him that sum: (1, 1).
- If he rolls a 3, there are two combinations: (1, 2) and (2, 1).
- If he rolls a 4, there are three combinations: (1, 3), (2, 2), and (3, 1).
- If he rolls a 5, there are four combinations: (1, 4), (2, 3), (3, 2), and (4, 1).
- If he rolls a 6, there are five combinations: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1).
- If he rolls a 7, there are six combinations: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1).
- If he rolls an 8, there are five combinations: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
- If he rolls a 9, there are four combinations: (3, 6), (4, 5), (5, 4), and (6, 3).
- If he rolls a 10, there are three combinations: (4, 6), (5, 5), and (6, 4).
- If he rolls an 11, there are two combinations: (5, 6) and (6, 5).
- If he rolls a 12, there's only one combination: (6, 6).
Adding up all these combinations, we get 1+2+3+4+5+6+5+4+3+2+1=36.
Since each roll of the dice gives us one combination, the maximum possible number of times Scott has thrown the dice is 36.
So, Scott must have thrown those dice a whopping 36 times without getting any repeated sums. That's quite the lucky streak!
To find the maximum possible number of times Scott has thrown the dice given that he has not rolled any sum twice, we need to consider the possibilities for the sums of the two dice.
Since Scott has not rolled any sum twice, each of the 11 possible sums from 2 to 12 must have been rolled once. The minimum number of times he would have thrown the dice to achieve this would be 11 rolls.
To show that this is indeed the maximum possible number of rolls, we need to consider the number of distinct sums that can be obtained from rolling the dice multiple times.
For each roll, the sum of the two dice falls within the range of 2 to 12. Since there are only 11 possible sums, once all of these sums have already been obtained, any subsequent roll will result in a sum that has been repeated before.
Hence, the maximum possible number of times Scott has thrown the dice is 11.