Question

A flag is to consist of six vertical stripes in yellow, green, blue, orange, brown, and red. It is not necessary to use all the colors. The same color may be used more than once. How many possible flags are there with no two adjacent stripes the same color?

Short answer

There are 18,750 possible flags.

Step-by-step solution

Understanding the Problem

We need to create flags consisting of six vertical stripes using the colors yellow, green, blue, orange, brown, and red. The key condition is that no two adjacent stripes can have the same color, and the same color can be reused across different stripes.

Choosing Colors for the First Stripe

The first stripe can be any one of the six colors: yellow, green, blue, orange, brown, or red. Thus, we have 6 choices for the first stripe.

Choosing Colors for the Subsequent Stripes

Each subsequent stripe must be different from the stripe immediately before it. Thus, for each of the remaining five stripes, there are 5 choices (since any color except the previous stripe's color can be used).

Calculating Total Possibilities

Multiply the number of choices for each stripe: The total number of possible flags is calculated as \(6 \times 5 \times 5 \times 5 \times 5 \times 5\).

Computing the Final Result

Calculate the expression from Step 4: \(6 \times 5^5 = 6 \times 3125 = 18750\). Therefore, there are 18,750 possible flags.

Key concepts

Permutations

Permutations refer to the arrangement of items or elements in a specific order. When solving problems involving arrangements, permutations play a critical role. In the context of our flag exercise, the arrangement of the colors is crucial because no two adjacent stripes can be the same color. Each stripe's color choice impacts the permissible colors for the adjacent stripe. Thus, despite having the flexibility to use any of the six colors for the first stripe, the sequence in which we order the colors for subsequent stripes matters significantly. Permutations become more nuanced when restrictions, such as the rule preventing identical adjacent colors, are applied. This changes the simple calculation from simply choosing six colors in a row to considering allowable sequences that meet the specified conditions. Understanding permutations helps us determine how to navigate these constraints when arranging items, which is foundational in combinatorics.

Combinatorial problems

Combinatorial problems encompass a range of mathematical problems focused on counting and arranging different sets of objects. In our exercise with flags, the problem revolves around counting all valid combinations of stripe arrangements where no two adjacent stripes share the same color. The broader field of combinatorics helps us find systematic ways to calculate these possibilities. The challenge often lies in adhering to specific rules while still accounting for all possible outcomes. Combinatorics explores various techniques, such as using permutations, combinations, or applications of the pigeonhole principle, to solve these problems efficiently. In complex problem settings, deciding which method to use becomes crucial for simplifying calculations and ensuring accuracy. Here, our focus on ensuring adjacent colors are different transforms a straightforward counting problem into an engaging combinatorial challenge.

Counting techniques

Counting techniques are fundamental to solving problems where you need to determine the number of ways something can occur. In the flag exercise, the objective is to correctly count all possible arrangements of stripes. Start with the first stripe, for which you have six choices. For each subsequent stripe, you get 5 options, since the rule requires that it be different from the one right before it. The counting is done by multiplying the choices together: after the first choice dictates the position, each of the remaining choices is multiplied (in this case, five times). This methodical approach of counting possibilities by multiplication—once any restraints are accounted for—ensures we can navigate complex arrangements. While the initial step is often straightforward, subsequent steps require analyzing how limitations affect subsequent choices. This systematic breakdown of choices is what makes counting techniques essential for providing accurate solutions to permutation and combination problems.