"Determining the number of subsets for a real-world situation"

Key Concepts about Sets and Subsets

• A set is a collection of distinct objects, considered as an object in its own right.
• The number of distinct subsets of a set is given by the formula:
  n = 2^m
  where m is the number of elements in the set.
• Each element can either be included or excluded from a subset, hence the power of 2.

Number of Subsets Example

• SAMPLE QUESTION:
  • Scott wants to order a pizza with specific toppings. The available toppings are:
  • bacon
  • mushrooms
  • olives
  • pepperoni
  • pineapple
  • sausage
  • spinach
  • tomatoes
  • Total toppings = 8.

Calculation of Subsets

• The total number of distinct subsets of 8 toppings is given by:
  2^8 = 256
• Important Condition: Scott cannot choose all toppings simultaneously; he can choose either some or none. Therefore, we need to exclude the scenario where all toppings are selected.

Final Calculation

• To find the number of ways Scott can choose some or none of the toppings:
  256 - 1 = 255

Conclusion

• The total number of ways Scott can choose the toppings, considering he can't choose all, is 255 distinct combinations.
• Understanding subsets helps in practical scenarios such as selecting pizza toppings, where options may be inclusive or exclusive based on certain rules.