A dichotomous key works by presenting two choices at each step to help classify items efficiently. This binary decision process is fundamental in organizing and identifying objects.

Facial expressions are clear and easily observable attributes on smiley faces. This binary trait makes them ideal for use in dichotomous keys.

A dichotomous key is designed to offer two alternatives at each decision point. This makes the classification process systematic and straightforward.

Starting with an obvious binary feature, like whether a smiley face is wearing a hat, simplifies the classification process. This initial decision helps narrow down the groups efficiently.

Happy vs. sad is a clear example of a binary classification system. It divides the group into two distinct categories based on facial expression.

Subtracting the 7 faces with glasses from the total of 12 leaves 5 without glasses. This basic subtraction problem reinforces simple arithmetic.

If 8 faces have big smiles, then the remaining 8 out of 16 have a smaller smile, which simplifies to 1/2. This reinforces understanding of fractions.

Two binary features produce 2 x 2 = 4 distinct categories. Each decision doubles the number of outcomes, reflecting exponential growth.

60% of 40 is 24, meaning 24 faces wear glasses; subtracting from 40 gives 16 faces without glasses. This question emphasizes percentage calculations.

5 out of 25 translates to 5/25, which simplifies to 1/5 or 20%. This conversion of fractions to percentages is crucial.

The maximum overlap is limited by the smaller percentage group, which is 30% in this case. Assuming all faces with extra eyes also have hats, the maximum is 30%.

With only blue and yellow options, there are 2 distinct background choices. This simple question reinforces understanding of binary choices.

Alternating sequences result in faces with hats at odd positions and without hats at even positions. The 10th face, being even, will not have a hat.

Multiplying 3 equations by 2 solutions per equation yields 6 smiley faces. This tests basic multiplication and counting skills.

After 3 steps, there are 2^3 = 8 categories, so dividing 32 by 8 yields 4 faces per category. This problem emphasizes division and exponential splits.

With three binary features, the total unique classifications are 2^3 = 8. This question reinforces concepts of combinatorics in binary outcomes.

Six binary steps create 2^6 = 64 distinct groups, meaning 64/64 equals 1 face per group. This highlights the exponential nature of binary division.

The probability of no error is calculated by multiplying the success rate for each decision: 0.95^3 ≈ 0.857 or about 86%. Multiplying independent probabilities shows the compounded effect.

Since 2^4 = 16 is not sufficient to classify 20 items, but 2^5 = 32 is enough, a minimum of 5 binary steps is required. This reinforces the concept of logarithmic growth in classification.

Calculating the overall percentage: 50% of faces wear hats and among these 70% wear glasses (0.5 * 70% = 35%), while 50% do not wear hats and 40% of these wear glasses (0.5 * 40% = 20%). Adding 35% and 20% gives 55%. This combines conditional probability with weighted averages.