Arrangements and Factorials: Solving a Combinatorial Puzzle

In this article, we delve into a fascinating problem in combinatorics. We explore the concept of permutations of a set and how adding members to a group affects the number of possible arrangements. Through a detailed solution, we will establish the initial number of boys and verify the solution.

Introduction to Factorials and Permutations

Factorials, denoted by n!, represent the product of all positive integers up to a given number n. Factorials play a crucial role in combinatorics, particularly in calculating the number of permutations or arrangements of a set of objects. A permutation refers to the arrangement of all the members of a set into some sequence or order.

The Problem at Hand

Suppose you have a group of n boys who can be arranged in a line in n! ways. When you add two more boys to the group, the number of possible arrangements increases by a factor of 420. The challenge is to determine the initial number of boys in the group, denoted by n.

Mathematical Formulation

To solve this problem, we start by expressing the number of arrangements:

n boys can be arranged in n! ways.

When two more boys are added, the number of boys becomes n 2, and the number of arrangements is (n 2)!.

According to the problem, the increase in the number of arrangements is by a factor of 420:

(n 2)! 420 times n!

Expanding (n 2)! in terms of n! gives:

(n 2)! (n 2) times (n 1) times n!

Substituting this into the equation:

(n 2)(n 1)n! 420 times n!

Assuming n! is not zero (which it isn't for n geq 0), we can divide both sides by n!:

(n 2)(n 1) 420

Expanding and simplifying:

n^2 3n 2 420

n^2 3n - 418 0

Applying the quadratic formula, a 1, b 3, c -418 to solve for n:

n frac{-b pm sqrt{b^2 - 4ac}}{2a}

frac{-3 pm sqrt{9 1672}}{2}

frac{-3 pm sqrt{1681}}{2}

frac{-3 pm 41}{2}

This gives us two solutions:

n frac{38}{2} 19

n frac{-44}{2} -22 (not a valid solution since n must be non-negative)

Verification

Thus, the only valid solution is n 19. To verify, let's check the permutations:

Original Problem Check:

For n 19:

19! text{ ways initial arrangement}

Adding two more boys, the number of arrangements is 21!:

frac{21!}{19!} 21 times 20 420 times frac{19!}{19!} 420

This verifies that the solution is correct.

Conclusion

Through this detailed exploration, we have solved the problem of determining the initial number of boys in a group given the increase in the number of possible arrangements after adding two more boys. The solution, n 19, stands as a testament to the power of combinatorial mathematics and factorials in solving such problems.

Additional Insights

Understanding the concept of permutations and factorials is crucial in combinatorics and can be applied to various real-world scenarios, from organizing schedules to analyzing statistical data. Whether you are a student or a professional in a field that requires logical reasoning and problem-solving skills, this knowledge can be invaluable.