To solve the problem described in the image, let's break it down step by step.

Problem Recap:

• There are 7 total bottles of car wash soap.
• 5 of these bottles contain orange soap.
• A club member randomly selects 4 bottles to pour into the car wash bucket.
• We need to find the probability that all 4 bottles selected contain orange soap.

Steps to Solve:

1. Total Combinations to Choose 4 Bottles: The total number of ways to select 4 bottles out of 7 is calculated using the combination formula: $\text{Total combinations} = \binom{7}{4} = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35$

2. Favorable Combinations (Selecting 4 Orange Bottles): The number of ways to choose 4 bottles out of the 5 orange soap bottles is: $\text{Favorable combinations} = \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5}{1} = 5$

3. Probability Calculation: The probability that all 4 selected bottles contain orange soap is: $\text{Probability} = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{5}{35} = \frac{1}{7} \approx 0.1429$ So, the probability is 0.1429 (rounded to four decimal places).

Final Answer:

The probability that all 4 selected bottles contain orange soap is 0.1429.

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Would you like more details or have any questions about this solution? Here are five related questions to explore:

1. How would the probability change if only 3 bottles were selected instead of 4?
2. What is the probability of selecting exactly 3 orange soap bottles out of the 4 selected?
3. How many ways are there to select 2 bottles containing orange soap and 2 containing another type?
4. If the number of total bottles were increased to 10, with 7 containing orange soap, what would be the probability of selecting 4 orange bottles?
5. How does the probability change if one additional bottle with orange soap is added to the original set?

Tip: When calculating probabilities involving combinations, always remember to clearly distinguish between the total number of possibilities and the number of favorable outcomes.