The Relationship Between Factorials and Array Arrangements

Oct 13, 2025 by miodeya.com 94 views

Introduction

In this problem, we are tasked with determining the number of coins that Dana has, given that she wants to display them in an array. The question asks us to identify which of the provided options (10, 16, 25, or 29 coins) would result in only 2 possible arrays for Dana to choose from.

The Concept of Factorials

To approach this problem, we need to understand the concept of factorials. A factorial of a number is the product of all positive integers less than or equal to that number. For example, the factorial of 5 (denoted as 5!) is equal to 5 × 4 × 3 × 2 × 1 = 120.

The Relationship Between Factorials and Array Arrangements

The number of possible arrays that can be formed from a given number of coins is related to the factorial of that number. Specifically, if we have n coins, the number of possible arrays is equal to the number of ways we can arrange those coins, which is given by the factorial of n (n!).

Analyzing the Options

Let's analyze each of the options provided:

Option A: 10 Coins

The number of possible arrays for 10 coins is 10! = 3,628,800. This is a very large number, and it's clear that there are many more than 2 possible arrays for 10 coins.

Option B: 16 Coins

The number of possible arrays for 16 coins is 16! = 20,922,789,888,000. Again, this is an extremely large number, and it's evident that there are many more than 2 possible arrays for 16 coins.

Option C: 25 Coins

The number of possible arrays for 25 coins is 25! = 15,511,210,043,330,985,984,000. As with the previous options, this is an enormous number, and it's clear that there are many more than 2 possible arrays for 25 coins.

Option D: 29 Coins

The number of possible arrays for 29 coins is 29! = 205,559,551,115,000,000,000,000. Once again, this is an incredibly large number, and it's evident that there are many more than 2 possible arrays for 29 coins.

The Correct Answer

However, we are looking for a number of coins that provides only 2 arrays for Dana to choose from. To find this number, we need to look for a number that has a factorial with a specific property.

The Property of Factorials

A factorial has a specific property when it is equal to the product of two consecutive integers. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120, and 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. Notice that 6! is equal to 6 × 5!, which means that 6! has a property that is related to the factorial of 5.

The Correct Number of Coins

Using this property, we can find the correct number of coins that provides only 2 arrays for Dana to choose from. We need to find a number that has a factorial with this property.

After analyzing the options, we find that the correct answer is:

Option B: 16 Coins

However, this is not the correct answer. The correct answer is actually Option D: 29 Coins is not the correct answer either.

The Correct Answer is actually 16 is not correct but 17 is not the answer either.

Q&A: Finding the Correct Number of Coins

Q: What is the problem about? A: The problem is about finding the number of coins that Dana has, given that she wants to display them in an array. We need to determine which of the provided options (10, 16, 25, or 29 coins) would result in only 2 possible arrays for Dana to choose from.

Q: What is the relationship between factorials and array arrangements? A: The number of possible arrays that can be formed from a given number of coins is related to the factorial of that number. Specifically, if we have n coins, the number of possible arrays is equal to the number of ways we can arrange those coins, which is given by the factorial of n (n!).

Q: How do we find the correct number of coins? A: We need to find a number that has a factorial with a specific property. A factorial has this property when it is equal to the product of two consecutive integers.

Q: What is the correct answer? A: After analyzing the options, we find that the correct answer is actually Option D: 29 Coins is not the correct answer. However, the correct answer is actually Option B: 16 Coins is not correct but Option D: 29 Coins is not the correct answer either.