Maryann rolls a 6-sided dice six times. How many ways can she roll the dice so that the sum of her numbers is 10?
Expressing the sum of the dice as ordered pairs ...
(4,6) , (6,4), (5,5)
So there are 3 ways
126
(1,1,1,1,1,5),(1,1,1,1,2,4),etc...
To find the number of ways Maryann can roll the dice so that the sum of her numbers is 10, let's break it down into smaller cases.
Case 1: Maryann gets six distinct numbers.
In this case, Maryann can only roll the dice in one specific way: 1, 2, 3, 4, 5, and 6. There is only one possible way to get a sum of 10 in this case.
Case 2: Maryann gets two identical numbers and four distinct numbers.
To get a sum of 10 in this case, Maryann needs to select one of her dice rolls to be the repeated number. She can choose this roll in 6 different ways (it can be any number from 1 to 6). Then, she needs to select four distinct numbers from the remaining 5 available numbers. She can do this in C(5,4) = 5 different ways. So, for each repeated number, there are 6 * 5 = 30 possible ways to get a sum of 10 in this case.
Case 3: Maryann gets two pairs of identical numbers.
In this case, Maryann needs to select two numbers to be the repeated pairs. She can choose these two numbers in C(6,2) = 15 different ways. Then, she needs to arrange the dice rolls to obtain a sum of 10. There are 6! / (2! * 2! * 2!) = 90 different ways to arrange the six dice rolls with two pairs of identical numbers. So, for each pair of numbers, there are 15 * 90 = 1350 possible ways to get a sum of 10 in this case.
Case 4: Maryann gets three identical numbers and three distinct numbers.
To get a sum of 10 in this case, Maryann first needs to select the number that will be repeated three times. She can choose this number in 6 different ways. Then, she needs to select three distinct numbers from the remaining 5 available numbers. She can do this in C(5,3) = 10 different ways. So, for each repeated number, there are 6 * 10 = 60 possible ways to get a sum of 10 in this case.
Case 5: Maryann gets four identical numbers and two distinct numbers.
Similar to case 4, Maryann can choose the number that will be repeated four times in 6 different ways. Then, she needs to select two distinct numbers from the remaining 5 available numbers, which can be done in C(5,2) = 10 different ways. So, for each repeated number, there are 6 * 10 = 60 possible ways to get a sum of 10 in this case.
Case 6: Maryann gets five identical numbers and one distinct number.
In this case, Maryann can choose the number that will be repeated five times in 6 different ways. There is only one distinct number left, so there is only one way to roll the dice to get a sum of 10 in this case.
Case 7: Maryann gets six identical numbers.
To get a sum of 10 in this case, Maryann needs to roll the dice with all six numbers equal to 10. There is only one possible way to do this.
Now, we can sum up the number of ways in each case to find the total number of ways Maryann can roll the dice to get a sum of 10:
1 + 30 + 1350 + 60 + 60 + 1 + 1 = 1503
Therefore, there are 1503 ways Maryann can roll the dice such that the sum of her numbers is 10.