Arranging Artistic Masterpieces: A Comprehensive Approach to Permutations with Constraints

Introduction

Consider an art gallery featuring a row of 10 paintings, with 5 by Picasso, 4 by Monet, and 1 by Turner. The challenge is to determine the number of different ways to display these paintings such that the works by each artist are kept together. This problem exemplifies a classic permutation issue and can be broken down into several manageable steps.p>

Step-by-Step Approach

Step 1: Grouping by Artist

The first step is to treat the paintings by each artist as single units. Therefore, we have:

5 Picasso paintings as one block 4 Monet paintings as one block 1 Turner painting as one block

This leaves us with 3 blocks to arrange.

Step 2: Arranging the Blocks

Since the 3 blocks can be arranged in different ways, we use the factorial of 3 to determine the permutations of these blocks:

(text{Number of ways to arrange blocks} 3! 3 times 2 times 1 6)

Step 3: Arranging Paintings Within Each Block

Within each block, we need to find the number of ways to arrange the paintings. For each artist, this can be calculated as follows:

The 5 Picasso paintings can be arranged in (5!) ways. The 4 Monet paintings can be arranged in (4!) ways. The 1 Turner painting can be arranged in (1!) way (which is 1).

Therefore:

(5! 120) (4! 24) (1! 1)

The total arrangements within the blocks are:

(5! times 4! times 1! 120 times 24 times 1 2880)

Step 4: Combining the Arrangements

To find the total number of different ways to display the paintings, we multiply the number of ways to arrange the blocks by the arrangements within each block:

(text{Total arrangements} 3! times 5! times 4! times 1! 6 times 120 times 24 times 1 17280)

Alternative Method

Alternatively, we can consider the paintings as three distinct groups, each with a specific number of paintings:

T Turner (1 painting) P Picasso (5 paintings together) M Monet (4 paintings together)

There are 6 possible permutations of these groups:

TPPPMMMM PPPTMMMM PPPMMTTT MTTTPPP MMMMTTT TTTTTMM

With 5! ways to arrange the Picasso paintings and 4! ways to arrange the Monet paintings, the total number of arrangements is:

(6 times 5! times 4! 6 times 120 times 24 17280)

Conclusion

By considering the paintings as blocks and then within the blocks, we can easily calculate the total number of arrangements. This problem not only demonstrates the principles of permutations but also highlights the importance of logical grouping and systematic computation in solving complex combinatorial problems.