Dimas wants to buy 10 fruits at the Fresh Fruit Shop consisting of mango, orange, and salak. If Dimas buys at least 2 fruits for each type, then the number of sample space members from the possible composition of the fruit purchased by Dimas is ....

A.1,440

D.260

B.720

E. 120

C.680

To determine the number of possible compositions of fruits Dimas can purchase, we need to find all combinations of mangoes (M), oranges (O), and salaks (S) where the total number of fruits is 10, and each type of fruit is bought in quantities of at least 2.

Let's denote the number of mangoes as \( M \), the number of oranges as \( O \), and the number of salaks as \( S \).

The problem conditions give us:
\[ M + O + S = 10 \]
\[ M \geq 2 \]
\[ O \geq 2 \]
\[ S \geq 2 \]

First, we redefine the variables to simplify the constraints by setting:
\[ M' = M - 2 \]
\[ O' = O - 2 \]
\[ S' = S - 2 \]

With these new variables \( M' \), \( O' \), and \( S' \), we now have:
\[ M' + O' + S' = 10 - 6 \]
\[ M' + O' + S' = 4 \]

Where \( M' \), \( O' \), and \( S' \) are non-negative integers (since \( M, O, S \geq 2 \)).

Now, we need to find the number of non-negative integer solutions to the equation \( M' + O' + S' = 4 \). This is a classic problem of combinations with repetition, also known as "stars and bars".

The formula to find the number of solutions in this case is given by:
\[ \binom{n + k - 1}{k - 1} \]

Where \( n = 4 \) (the total we need to sum to) and \( k = 3 \) (the number of variables: \( M', O', S' \)).

So, the number of solutions is:
\[ \binom{4 + 3 - 1}{3 - 1} = \binom{6}{2} \]

Calculating the binomial coefficient:
\[ \binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5}{2 \times 1} = 15 \]

Thus, the number of possible compositions of fruits Dimas can purchase is:
\[ \boxed{15} \]

None of the options (A. 1,440, B. 720, C. 680, D. 260, E. 120) are correct as per the given solution.

Would you like further details on this problem or have any other questions? Here are some related questions for further exploration:

1. How do you calculate combinations with repetition?
2. What are binomial coefficients and how are they used?
3. How can we generalize the stars and bars method for different sums?
4. What if the constraints on minimum fruits were different, how would the problem change?
5. Can you explain the logic behind the transformation of variables in such problems?
6. What are some other applications of the stars and bars method in combinatorics?
7. How would this problem be solved if Dimas wanted to buy exactly 10 fruits but had different constraints (e.g., at least 1 of each type)?
8. How would this problem change if there were 4 types of fruits instead of 3?

**Tip:** When dealing with combinatorial problems, breaking down constraints and transforming variables can greatly simplify the process.

Learn how to calculate the number of possible compositions of fruits Dimas can purchase under minimum purchase conditions using combinatorial methods. This problem involves solving equations with constraints and applying the Stars and Bars Theorem. Suitable for students in Grades 10-12.

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