A store stocks 8 different kinds of batteris, one of which is AA, with at least 28 packages of each kind how many ways can a purchase of 28 packages be made?

The correct answer and explanation is:

We are given:

  • A store stocks 8 different kinds of batteries, including AA.
  • Each kind has at least 28 packages, so availability is not a limiting factor.
  • A customer wants to buy exactly 28 packages in total.
  • We are to find how many different combinations of 28 packages can be purchased, choosing from these 8 types.

🔢 Step-by-Step Breakdown

Let the 8 types of batteries be represented by variables: x1+x2+x3+x4+x5+x6+x7+x8=28x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 = 28

Where:

  • xix_i = number of packages of the ii-th type of battery purchased.
  • Each xi≥0x_i \geq 0 (non-negative integers).

This is a “stars and bars” combinatorics problem — the number of non-negative integer solutions to the equation: x1+x2+⋯+xk=nx_1 + x_2 + \dots + x_k = n

is given by: (n+k−1k−1)\binom{n + k – 1}{k – 1}

Here, n=28n = 28 and k=8k = 8: (28+8−18−1)=(357)\binom{28 + 8 – 1}{8 – 1} = \binom{35}{7}

✅ Final Answer:

(357)=6, ⁣724, ⁣520\boxed{\binom{35}{7} = 6,\!724,\!520}


📘 Explanation (300 words)

This problem involves choosing how to distribute 28 indistinguishable packages of batteries among 8 different kinds. This is a classic example of a combinations with repetition problem in combinatorics.

Each different distribution (like 4 AA, 3 AAA, 5 D, etc.) is a valid way to purchase the 28 packages. We’re not concerned with the order of purchase or the package appearance, just how the total is split across the 8 kinds.

Because the store has at least 28 of each type, we don’t have to worry about running out of a specific kind. That’s important — it lets us assume unlimited supply, and apply the stars and bars method.

Think of the 28 packages as 28 stars (*). We want to divide these stars into 8 groups (battery types), using 7 bars (|) to separate them. For example:

****|**||*****|*|*******||***

This represents a possible distribution of the 28 packages among the 8 types.

The number of ways to place these 7 dividers among the 35 total slots (28 stars + 7 bars) is: (357)\binom{35}{7}

This is calculated as: (357)=35!7!⋅28!=6, ⁣724, ⁣520\binom{35}{7} = \frac{35!}{7! \cdot 28!} = \boxed{6,\!724,\!520}

Thus, 6,724,520 different purchase combinations are possible.

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