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Chapter 3: Problem 21

A lady wishes to color her fingernails on one hand using at most two of the colors red, yellow, and blue. How many ways can she do this?

Short Answer

Expert verified
The total number of ways is 21.

Step by step solution

01

Identifying Color Combinations

We need to determine all possible combinations of choosing at most two colors from the colors available: red, yellow, and blue. The combinations are: picking only red, picking only yellow, picking only blue, picking both red and yellow, picking both red and blue, and picking both yellow and blue.
02

Calculate Ways for each Combination

For each combination, calculate the number of ways to paint the nails on one hand (typically 5 nails). - One color only: Choose any 1 color from the 3 and apply it to all 5 nails. - Two colors: Choose any 2 colors from the 3, and distribute them on 5 nails. For example, if you choose red and yellow, apply any combination like all red, four red one yellow, three red two yellow, etc.
03

Calculating Single Color Options

For each single color, there is exactly one way to color all 5 nails. Therefore, with 3 colors available (red, yellow, blue), we have 3 combinations.
04

Calculating Two Colors Options

For two colors, calculate the number of combinations for each scenario: - Choose any 2 colors out of 3, then apply them in combinations over 5 nails using combinations of 5 things taken so as to separate them in distinct groups (like determining all subsets of combination). - For each pair, the nails can be arranged by determining all possible strings only comprising the chosen colors like permutation and combination involve choosing places for each color.
05

Counting Two Color Options

For simplicity, assuming nails are to be colored distinctly: - Red & Yellow: Choose ways to apply 5 spots (one of 0-5) with either R or Y can be taken as involving no constraint over specific combinations. - Similarly calculate for Red & Blue and Yellow & Blue pairs totaling such cases.
06

Total Calculation

Sum the total combinations: 3 options for one-color cases plus the summed unique distribution possibilities, providing solution in additive segments to measure valid pairwise attempts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutations and Combinations
The concept of permutations and combinations forms the basis of solving the exercise with nail painting. Let鈥檚 break it down.
  • Permutations involve arranging elements in a sequence. It is about finding the number of ways to arrange a set of items where order matters.
  • Combinations focus on selecting items from a group where order does not matter. For example, choosing 2 colors from red, yellow, and blue is strictly a combination problem since the order of picking does not change the outcome.
For this nail color exercise, we calculate the combinations for choosing colors, then use permutations to arrange the selected color(s) across the nails. This involves selecting one or two colors and figuring out how to arrange them on five nails, either by maintaining a color or mixing two colors.
Color Combinations
Choosing the right combinations of colors is crucial in solving coloring problems like this. The key is to explore all possibilities when selecting colors:
  • Picking a single color, such as just red, means all five nails would be the same color.
  • When opting for two colors, say red and yellow, you're looking at various ways to mix these across five nails. This can include having three nails red and two nails yellow, or any other combination where the two colors are present.
Each selection opens up different possibilities, leading to an overall goal of determining how many unique ways the colors can be arranged on the nails. Recognizing all eligible combinations ensures all scenarios are covered.
Distinct Arrangements
Considering distinct arrangements, especially in a coloring scenario, expands on finding unique design options. When we aim for distinct arrangements:
  • With a single color, the approach is straightforward as simply all the nails are colored the same.
  • Using two colors, distinct arrangements become more involved. Each pair of colors chosen can lead to a variety of combinations, like four nails of one color and one of another or alternating colors on nails.
In this exercise, achieving distinct arrangements means evaluating all methods of distributing your chosen colors’ spots on the nails. These arrangements enable achieving uniqueness in design, implying different sequences and patterns are formed with every pairing of colors.

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Most popular questions from this chapter

Let \(j\) and \(n\) be positive integers, with \(j \leq n\). An experiment consists of choosing, at random, a \(j\) -tuple of positive integers whose sum is at most \(n\). (a) Find the size of the sample space. Hint: Consider \(n\) indistinguishable balls placed in a row. Place \(j\) markers between consecutive pairs of balls, with no two markers between the same pair of balls. (We also allow one of the \(n\) markers to be placed at the end of the row of balls.) Show that there is a \(1-1\) correspondence between the set of possible positions for the markers and the set of \(j\) -tuples whose size we are trying to count. (b) Find the probability that the \(j\) -tuple selected contains at least one 1 .

A certain state has license plates showing three numbers and three letters. How many different license plates are possible (a) if the numbers must come before the letters? (b) if there is no restriction on where the letters and numbers appear?

Let \(X\) denote a particular process that produces elements of \(S_{n},\) and let \(U\) denote the uniform process. Let the distribution functions of these processes be denoted by \(f_{X}\) and \(u,\) respectively. Show that the variation distance \(\left\|f_{X}-u\right\|\) is equal to $$ \max _{T \subset S_{n}} \sum_{\pi \in T}\left(f_{X}(\pi)-u(\pi)\right) $$ Hint: Write the permutations in \(S_{n}\) in decreasing order of the difference \(f_{X}(\pi)-u(\pi)\).

There are three different routes connecting city A to city B. How many ways can a round trip be made from \(A\) to \(B\) and back? How many ways if it is desired to take a different route on the way back?

Suppose that on planet Zorg a year has \(n\) days, and that the lifeforms there are equally likely to have hatched on any day of the year. We would like to estimate \(d\), which is the minimum number of lifeforms needed so that the probability of at least two sharing a birthday exceeds \(1 / 2\). (a) In Example 3.3 , it was shown that in a set of \(d\) lifeforms, the probability that no two life forms share a birthday is $$ \frac{(n)_{d}}{n^{d}} $$ where \((n)_{d}=(n)(n-1) \cdots(n-d+1)\). Thus, we would like to set this equal to \(1 / 2\) and solve for \(d\). (b) Using Stirling's Formula, show that $$ \frac{(n)_{d}}{n^{d}} \sim\left(1+\frac{d}{n-d}\right)^{n-d+1 / 2} e^{-d} $$ (c) Now take the logarithm of the right-hand expression, and use the fact that for small values of \(x\), we have $$ \log (1+x) \sim x-\frac{x^{2}}{2} $$ (We are implicitly using the fact that \(d\) is of smaller order of magnitude than \(n\). We will also use this fact in part (d).) (d) Set the expression found in part (c) equal to \(-\log (2),\) and solve for \(d\) as a function of \(n\), thereby showing that $$ d \sim \sqrt{2(\log 2) n} $$ Hint: If all three summands in the expression found in part (b) are used, one obtains a cubic equation in \(d\). If the smallest of the three terms is thrown away, one obtains a quadratic equation in \(d\). (e) Use a computer to calculate the exact values of \(d\) for various values of \(n\). Compare these values with the approximate values obtained by using the answer to part \(\mathrm{d}\) ).
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