To determine how many distinct products can be formed by selecting two different integers from the given set [tex]\(\{-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}\)[/tex], we will follow a systematic approach:
1. Identify all pairs of different integers:
- We need to consider all unique pairs [tex]\((a, b)\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are different elements of the set.
2. Compute the product for each pair:
- For each unique pair [tex]\((a, b)\)[/tex], calculate the product [tex]\(a \times b\)[/tex].
3. Record distinct products:
- Use a method to record products without duplicating any values.
4. Count the distinct products:
- Finally, count the number of unique products.
### Detailed Steps:
1. List of Pairs:
- We need combinations of the set taken 2 at a time. For example, some pairs include [tex]\((-6, -5)\)[/tex], [tex]\((-6, -4)\)[/tex], [tex]\((-6, 5)\)[/tex], and so on.
2. Calculate Products:
- For each pair, compute the product. For example:
- [tex]\((-6) \times (-5) = 30\)[/tex]
- [tex]\((-6) \times (-4) = 24\)[/tex]
- [tex]\((-6) \times 5 = -30\)[/tex]
- And so forth for all possible pairs.
3. Distinct Products Set:
- Create a set to store the products. A set inherently avoids duplicate values. This means every time a product is added to the set, it will only be stored if it is not already present.
4. Products Computation:
- After calculating and storing in a set, the products might look like:
- [tex]\(30, 24, -30, \ldots, 0, 2, 3, -9, \ldots\)[/tex]
- Continue this process until all pairs are evaluated.
5. Count Distinct Products:
- After processing all pairs, the set will contain all distinct products.
- The number of distinct products is the size of the set.
When completed, we find that there are 31 distinct products that can be formed using pairs of integers from the given set. Here is the list of those distinct products:
[tex]\[ 0, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 20, 24, 30, -9, -30, -25, -24, -20, -18, -16, -15, -12, -10, -2, -8, -6, -5, -4, -3, -1 \][/tex]
So, the number of distinct products is [tex]\(\boxed{31}\)[/tex].
