Final answer:
There are 108 different lunch specials available at the local diner. For the 6-bit strings question, there are 57 possible strings that have two or more 1’s.
Step-by-step explanation:
To determine how many different choices there are for a lunch special at the local diner, we need to consider the combinations of beverages, sandwiches, and sides. The diner offers 4 different beverages, 7 different sandwiches (with 2 sandwiches that can be ordered either toasted or cold), and 3 different sides. Counting the sandwich options, we have 5 sandwiches that can only be ordered cold and 2 sandwiches that can be ordered in 2 different ways, either toasted or cold. This gives us 5 + (2 × 2) = 9 sandwich options. To get the total number of combinations, we multiply the options for each choice together: 4 (beverages) × 9 (sandwiches) × 3 (sides) = 108 different lunch specials.
Regarding the 6-bit strings, these are binary sequences containing six positions that can be either 0 or 1. To find the number of 6-bit strings with two or more 1’s, we first consider the total number of possible 6-bit strings, which is 26 = 64. From this total, we need to subtract the number of strings that do not meet the condition, which are the strings with 0 or 1 instances of '1'. There is only 1 string with no '1's (all zeros), and there are 6 strings with exactly one '1' (each position can be the location of the single '1'). Subtracting these from the total gives us 64 - 1 - 6 = 57 strings with two or more 1’s.