Calculating the Number of Ways to Roll a Full House

Oct 13, 2025 by wbrum.org 2 views

Introduction

In the world of probability and statistics, understanding the behavior of random events is crucial. One such event is rolling dice, which has been a popular form of entertainment for centuries. In this article, we will delve into the problem of finding the number of distinct ways Milo can roll a full house with five fair dice, each having a different number of sides.

What is a Full House?

A full house is a hand or roll in which three dice show one number, and the other two dice show another number. For example, if Milo rolls three 4s and two 6s, it is considered a full house. The objective is to find the number of distinct ways Milo can achieve this.

Calculating the Number of Ways to Roll a Full House

To calculate the number of ways to roll a full house, we need to consider the following steps:

Choosing the Number for the Three Dice: The first step is to choose the number that will appear on the three dice. Since each die has a different number of sides, we need to consider the number of sides for each die. The first die has 4 sides, the second die has 6 sides, the third die has 8 sides, the fourth die has 12 sides, and the fifth die has 20 sides.
Choosing the Number for the Two Dice: The second step is to choose the number that will appear on the two dice. This number must be different from the number chosen in the first step.
Calculating the Number of Ways to Roll the Three Dice: Once we have chosen the number for the three dice, we need to calculate the number of ways to roll this number on the three dice. This can be done using the formula for combinations, which is given by:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of sides on the die, and k is the number of sides that show the chosen number.
Calculating the Number of Ways to Roll the Two Dice: Similarly, once we have chosen the number for the two dice, we need to calculate the number of ways to roll this number on the two dice.

Calculating the Total Number of Ways to Roll a Full House

To calculate the total number of ways to roll a full house, we need to multiply the number of ways to choose the number for the three dice, the number of ways to choose the number for the two dice, the number of ways to roll the three dice, and the number of ways to roll the two dice.

Step 1: Choosing the Number for the Three Dice

The first step is to choose the number that will appear on the three dice. Since each die has a different number of sides, we need to consider the number of sides for each die. The first die has 4 sides, the second die has 6 sides, the third die has 8 sides, the fourth die has 12 sides, and the fifth die has 20 sides.

Let's consider the number of ways to choose the number for the three dice. We can choose any number from 1 to 4 for the first die, any number from 1 to 6 for the second die, any number from 1 to 8 for the third die, any number from 1 to 12 for the fourth die, and any number from 1 to 20 for the fifth die.

import math
die1_sides = 4
die2_sides = 6
die3_sides = 8
die4_sides = 12
die5_sides = 20
num_ways_die1 = die1_sides
num_ways_die2 = die2_sides
num_ways_die3 = die3_sides
num_ways_die4 = die4_sides
num_ways_die5 = die5_sides

Step 2: Choosing the Number for the Two Dice

The second step is to choose the number that will appear on the two dice. This number must be different from the number chosen in the first step.

Let's consider the number of ways to choose the number for the two dice. We can choose any number from 1 to 3 for the first die, any number from 1 to 5 for the second die, any number from 1 to 7 for the third die, any number from 1 to 11 for the fourth die, and any number from 1 to 19 for the fifth die.

# Calculate the number of ways to choose the number for the two dice
num_ways_die1_2 = die1_sides - 1
num_ways_die2_2 = die2_sides - 1
num_ways_die3_2 = die3_sides - 1
num_ways_die4_2 = die4_sides - 1
num_ways_die5_2 = die5_sides - 1

Step 3: Calculating the Number of Ways to Roll the Three Dice

Once we have chosen the number for the three dice, we need to calculate the number of ways to roll this number on the three dice. This can be done using the formula for combinations, which is given by:

C(n, k) = n! / (k!(n-k)!)

where n is the total number of sides on the die, and k is the number of sides that show the chosen number.

# Calculate the number of ways to roll the three dice
num_ways_roll_die1 = math.comb(die1_sides, 3)
num_ways_roll_die2 = math.comb(die2_sides, 3)
num_ways_roll_die3 = math.comb(die3_sides, 3)
num_ways_roll_die4 = math.comb(die4_sides, 3)
num_ways_roll_die5 = math.comb(die5_sides, 3)

Step 4: Calculating the Number of Ways to Roll the Two Dice

Similarly, once we have chosen the number for the two dice, we need to calculate the number of ways to roll this number on the two dice.

# Calculate the number of ways to roll the two dice
num_ways_roll_die1_2 = math.comb(die1_sides - 1, 2)
num_ways_roll_die2_2 = math.comb(die2_sides - 1, 2)
num_ways_roll_die3_2 = math.comb(die3_sides - 1, 2)
num_ways_roll_die4_2 = math.comb(die4_sides - 1, 2)
num_ways_roll_die5_2 = math.comb(die5_sides - 1, 2)

Calculating the Total Number of Ways to Roll a Full House

# Calculate the total number of ways to roll a full house
total_num_ways = (num_ways_die1 * num_ways_die2 * num_ways_die3 * num_ways_die4 * num_ways_die5) * \
                 (num_ways_die1_2 * num_ways_die2_2 * num_ways_die3_2 * num_ways_die4_2 * num_ways_die5_2) * \
                 (num_ways_roll_die1 * num_ways_roll_die2 * num_ways_roll_die3 * num_ways_roll_die4 * num_ways_roll_die5) * \
                 (num_ways_roll_die1_2 * num_ways_roll_die2_2 * num_ways_roll_die3_2 * num_ways_roll_die4_2 * num_ways_roll_die5_2)

Conclusion

In this article, we have calculated the total number of ways Milo can roll a full house with five fair dice, each having a different number of sides. The total number of ways is given by the product of the number of ways to choose the number for the three dice, the number of ways to choose the number for the two dice, the number of ways to roll the three dice, and the number of ways to roll the two dice.

Final Answer

Introduction

In our previous article, we explored the problem of finding the number of distinct ways Milo can roll a full house with five fair dice, each having a different number of sides. In this article, we will answer some frequently asked questions related to this problem.

Q: What is a full house in dice rolls?

A: A full house is a hand or roll in which three dice show one number, and the other two dice show another number. For example, if Milo rolls three 4s and two 6s, it is considered a full house.

Q: How many distinct ways can Milo roll a full house with five fair dice?

A: To calculate the total number of ways to roll a full house, we need to multiply the number of ways to choose the number for the three dice, the number of ways to choose the number for the two dice, the number of ways to roll the three dice, and the number of ways to roll the two dice.

Q: What is the formula for calculating the number of ways to roll a full house?

A: The formula for calculating the total number of ways to roll a full house is given by:

total_num_ways = (num_ways_die1 * num_ways_die2 * num_ways_die3 * num_ways_die4 * num_ways_die5) * \
                 (num_ways_die1_2 * num_ways_die2_2 * num_ways_die3_2 * num_ways_die4_2 * num_ways_die5_2) * \
                 (num_ways_roll_die1 * num_ways_roll_die2 * num_ways_roll_die3 * num_ways_roll_die4 * num_ways_roll_die5) * \
                 (num_ways_roll_die1_2 * num_ways_roll_die2_2 * num_ways_roll_die3_2 * num_ways_roll_die4_2 * num_ways_roll_die5_2)

Q: What are the values of num_ways_die1, num_ways_die2, num_ways_die3, num_ways_die4, and num_ways_die5?

A: The values of num_ways_die1, num_ways_die2, num_ways_die3, num_ways_die4, and num_ways_die5 are given by:

num_ways_die1 = die1_sides
num_ways_die2 = die2_sides
num_ways_die3 = die3_sides
num_ways_die4 = die4_sides
num_ways_die5 = die5_sides

Q: What are the values of num_ways_die1_2, num_ways_die2_2, num_ways_die3_2, num_ways_die4_2, and num_ways_die5_2?

A: The values of num_ways_die1_2, num_ways_die2_2, num_ways_die3_2, num_ways_die4_2, and num_ways_die5_2 are given by:

num_ways_die1_2 = die1_sides - 1
num_ways_die2_2 = die2_sides - 1
num_ways_die3_2 = die3_sides - 1
num_ways_die4_2 = die4_sides - 1
num_ways_die5_2 = die5_sides - 1

Q: What are the values of num_ways_roll_die1, num_ways_roll_die2, num_ways_roll_die3, num_ways_roll_die4, and num_ways_roll_die5?

A: The values of num_ways_roll_die1, num_ways_roll_die2, num_ways_roll_die3, num_ways_roll_die4, and num_ways_roll_die5 are given by:

num_ways_roll_die1 = math.comb(die1_sides, 3)
num_ways_roll_die2 = math.comb(die2_sides, 3)
num_ways_roll_die3 = math.comb(die3_sides, 3)
num_ways_roll_die4 = math.comb(die4_sides, 3)
num_ways_roll_die5 = math.comb(die5_sides, 3)

Q: What are the values of num_ways_roll_die1_2, num_ways_roll_die2_2, num_ways_roll_die3_2, num_ways_roll_die4_2, and num_ways_roll_die5_2?

A: The values of num_ways_roll_die1_2, num_ways_roll_die2_2, num_ways_roll_die3_2, num_ways_roll_die4_2, and num_ways_roll_die5_2 are given by:

num_ways_roll_die1_2 = math.comb(die1_sides - 1, 2)
num_ways_roll_die2_2 = math.comb(die2_sides - 1, 2)
num_ways_roll_die3_2 = math.comb(die3_sides - 1, 2)
num_ways_roll_die4_2 = math.comb(die4_sides - 1, 2)
num_ways_roll_die5_2 = math.comb(die5_sides - 1, 2)

Conclusion

In this article, we have answered some frequently asked questions related to the problem of finding the number of distinct ways Milo can roll a full house with five fair dice, each having a different number of sides. We hope that this article has provided a clear understanding of the problem and its solution.