To solve this problem, we need to calculate the probability that fewer than two oranges are chosen when 7 fruits are selected from a total of 8 apples and 10 oranges.

Step 1: Total possible selections

The total number of ways to select 7 fruits from the 18 fruits (8 apples + 10 oranges) is given by the combination formula:

$\binom{18}{7} = \frac{18!}{7!(18-7)!} = \binom{18}{7}$

Step 2: Defining "fewer than two oranges"

The phrase "fewer than two oranges" means we can select:

• 0 oranges (which means 7 apples),
• 1 orange (which means 6 apples).

We need to calculate the number of ways to select 0 oranges and 1 orange.

Step 3: Calculate selections for 0 oranges

If 0 oranges are chosen, all 7 fruits must be apples. The number of ways to choose 7 apples from the 8 apples is:

$\binom{8}{7} = \frac{8!}{7!(8-7)!} = 8$

Step 4: Calculate selections for 1 orange

If 1 orange is chosen, then 6 apples must be chosen. The number of ways to choose 1 orange from the 10 oranges and 6 apples from the 8 apples is:

$\binom{10}{1} \times \binom{8}{6} = 10 \times \frac{8!}{6!(8-6)!} = 10 \times 28 = 280$

Step 5: Total favorable outcomes

The total number of favorable outcomes (either 0 oranges or 1 orange) is:

$\text{Favorable outcomes} = 8 + 280 = 288$

Step 6: Total possible outcomes

The total number of ways to choose 7 fruits from 18 is:

$\binom{18}{7} = 31824$

Step 7: Probability calculation

The probability is the ratio of favorable outcomes to total possible outcomes:

$P(\text{fewer than two oranges}) = \frac{288}{31824} = \frac{1}{110.5} \approx 0.009$

Thus, the probability of choosing fewer than two oranges is approximately $0.009$, or about 0.9%.

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Would you like more details, or have any questions?

Here are 5 related questions to explore this concept further:

1. How would the probability change if you had more apples and fewer oranges?
2. What is the probability of choosing exactly 2 oranges?
3. Can you derive the total number of ways to select any 7 fruits from 18 using factorials?
4. What is the general formula for calculating probabilities in combinatorics-based problems like this?
5. How does changing the number of fruits selected (say, from 7 to 5) impact the results?

Tip: Always confirm whether the conditions of a problem imply "at least" or "fewer than" conditions for a better understanding of the probability problem.