A grocery store manager took inventory of all the fruit to decide what needed to be donated. Of the 31 pieces of fruit deemed "overripe," 4 were pears and 17 were peaches.

If the manager randomly selected 6 of the overripe pieces for the first donation package, what is the probability that 3 of the chosen fruits are pears and 3 are peaches?

*Write your* *answer as a decimal rounded to four decimal* *places.*




To solve this problem, we need to calculate the probability that 3 pears and 3 peaches are selected from the overripe fruits. Here's the breakdown:

### Step 1: Total Fruits and Selections
- Total number of overripe fruits = \( 31 \)
- Number of pears = \( 4 \)
- Number of peaches = \( 17 \)
- Total number of fruits to select = \( 6 \)
- We want to select 3 pears and 3 peaches.

### Step 2: Use of Combinations
We will use the combination formula, denoted as \( C(n, k) \), which is:

\[
C(n, k) = \frac{n!}{k!(n - k)!}
\]

Where \( n \) is the total number of items, and \( k \) is the number of items to choose.

#### Step 3: Calculate the Number of Ways to Select the Fruits
- Number of ways to choose 3 pears from 4 pears:
  \[
  C(4, 3) = \frac{4!}{3!(4-3)!} = \frac{4!}{3!1!} = 4
  \]
  
- Number of ways to choose 3 peaches from 17 peaches:
  \[
  C(17, 3) = \frac{17!}{3!(17-3)!} = \frac{17!}{3!14!} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 680
  \]

- Total number of ways to select 6 fruits from 31 fruits:
  \[
  C(31, 6) = \frac{31!}{6!(31-6)!} = \frac{31 \times 30 \times 29 \times 28 \times 27 \times 26}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = 736,281
  \]

### Step 4: Calculate the Probability
The probability is the ratio of the number of favorable outcomes to the total possible outcomes:

\[
\text{Probability} = \frac{C(4, 3) \times C(17, 3)}{C(31, 6)} = \frac{4 \times 680}{736,281} = \frac{2,720}{736,281} \approx 0.0037
\]

Thus, the probability that exactly 3 pears and 3 peaches are selected is approximately \( \boxed{0.0037} \).

---

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. What if 4 peaches and 2 pears were selected instead?
2. How would the probability change if there were 6 pears and 17 peaches?
3. What is the probability of selecting 2 pears and 4 peaches?
4. How many different ways can 4 peaches be selected from the group of 17 peaches?
5. What if we selected 7 fruits instead of 6?

**Tip**: In probability, always ensure that you count the favorable outcomes and the total outcomes correctly using combinations when dealing with selections!

A grocery store manager took inventory of all the fruit to decide what needed to be donated. Of the 31 pieces of fruit deemed 'overripe,' 4 were pears and 17 were peaches. If the manager randomly selected 6 of the overripe pieces for the first donation package, what is the probability that 3 of the chosen fruits are pears and 3 are peaches? Write your answer as a decimal rounded to four decimal places.

Explore how to calculate the probability of selecting 3 pears and 3 peaches from a total of 6 overripe fruits. This problem involves key combinatorics concepts and uses the combination formula to determine the favorable outcomes. Detailed step-by-step solution included. Suitable for students in Grades 10-12.

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