Respuesta :
The number of different ways for each case is given as follows:
a) 1,024 ways.
b) 1,043,456 ways.
c) 1,073,739,264 ways.
What is the Fundamental Counting Theorem?
The Fundamental Counting Theorem states that if there are n independent trials, each with [tex]n_1, n_2, \cdots, n_n[/tex] possible results, the total number of outcomes is calculated by the multiplication of the number of outcomes for each trial as presented as follows:
[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]
For item a, we have that for each of the 10 toppings, there are 2 options, insert or not, hence the total number of outcomes is of:
2^10 = 1024 ways.
For item b, we have that:
- Each of Grogg and Lizzie has 1024 possible outcomes.
- The cases with one or more toppings in common have to be subtracted. There are 10 ways to choose a common topping, and 2^9 = 512 different ways to choose the remaining toppings.
Hence the number of ways is given as follows:
1024 x 1024 - 10 x 512 = 1,043,456
For item c, we have that:
- Each of Grogg, Lizzie and Winnie has 1024 possible outcomes.
- For Grogg and Lizzie, there are 1024 x 1024 possible outcomes.
- The cases in which Winnie has a topping in common with them have to be subtracted.
- There are 10 common toppings that Winnie can choose, with 2^8 = 256 possible combinations.
Hence the number of ways is given as follows:
1024 x 1024 x 1024 - 10 x 256 = 1,073,739,264
More can be learned about the Fundamental Counting Theorem at https://brainly.com/question/15878751
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