There are eight apples and ten oranges. Seven fruits are chosen from these fruits. Find the probability that fewer than two oranges are chosed from those fruits

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.

There are eight apples and ten oranges. Seven fruits are chosen from these fruits. Find the probability that fewer than two oranges are chosen from those fruits.

This problem involves calculating the probability of selecting fewer than two oranges when choosing 7 fruits from a group of 8 apples and 10 oranges. It uses key concepts in combinatorics and probability, including the combination formula and basic probability theorem. Suitable for students in Grades 9-12.

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