Miyu is giving out $8$ identical chocolates to her $5$ friends, including Dhruv. All possible distributions are equally likely. What is the probability that Dhruv gets exactly $1$ chocolate?

To solve this problem, we need to determine the probability that Dhruv gets exactly 1 chocolate out of the 8 chocolates that Miyu is distributing to her 5 friends.

### Step 1: Total number of distributions
We are distributing 8 identical chocolates among 5 friends. The number of ways to distribute $ n $ identical items among $ k $ distinct recipients can be computed using the "stars and bars" theorem. The formula is:

\[
\text{Number of distributions} = \binom{n+k-1}{k-1}
\]

For our problem:
- $ n = 8 $ (number of chocolates)
- $ k = 5 $ (number of friends)

So, the total number of distributions is:

\[
\text{Total number of distributions} = \binom{8+5-1}{5-1} = \binom{12}{4}
\]

Now, calculate $ \binom{12}{4} $:

\[
\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} = \frac{11880}{24} = 495
\]

### Step 2: Number of favorable distributions (Dhruv gets exactly 1 chocolate)
If Dhruv gets exactly 1 chocolate, we are left with 7 chocolates to distribute among the remaining 4 friends. The number of ways to distribute 7 chocolates among 4 friends is:

\[
\text{Number of favorable distributions} = \binom{7+4-1}{4-1} = \binom{10}{3}
\]

Now, calculate $ \binom{10}{3} $:

\[
\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120
\]

### Step 3: Calculate the probability
The probability that Dhruv gets exactly 1 chocolate is the ratio of the number of favorable distributions to the total number of distributions:

\[
\text{Probability} = \frac{\text{Number of favorable distributions}}{\text{Total number of distributions}} = \frac{120}{495}
\]

Simplify the fraction:

\[
\text{Probability} = \frac{120}{495} = \frac{24}{99} = \frac{8}{33}
\]

Thus, the probability that Dhruv gets exactly 1 chocolate is $ \boxed{\frac{8}{33}} $.

Would you like to know more details or ask any questions? Here are 5 additional questions related to this topic:

1. What if Dhruv must receive at least 2 chocolates? How does the probability change?
2. How would the probability be affected if there were 6 friends instead of 5?
3. Can you derive the general formula for the probability that a specific friend receives exactly $ k $ chocolates?
4. What if the chocolates were not identical? How would this change the calculation?
5. How does the stars and bars method work in combinatorics?

**Tip:** When working with combinatorial problems, identifying the correct use of combinatorial formulas (like "stars and bars") is key to finding the solution efficiently.

Explore a probability problem where Miyu distributes 8 identical chocolates among her 5 friends, including Dhruv. Learn how to calculate the probability that Dhruv gets exactly 1 chocolate using combinatorial methods. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation