using the fundamental counting principle with three or more groups of items
There are three highways from city A to city B, two highways from city B to city C,and four highways from city c to city D. How many different highway routes are there from city A to city D?
I see 3*2*4= ?
Those are direct paths. If you allow going back and forth, such as A to b, then B to C, then C to B (on another path), the B to C (on anohter path), well the count increases considerably.
To use the fundamental counting principle, you need to multiply the number of choices in each group. In this case, we have three groups: highways from A to B, B to C, and C to D.
First, let's calculate the number of choices in each group:
1. Highways from A to B: There are three highways from city A to city B, so there are three choices.
2. Highways from B to C: There are two highways from city B to city C, so there are two choices.
3. Highways from C to D: There are four highways from city C to city D, so there are four choices.
Now, multiply the number of choices in each group:
3 highways from A to B * 2 highways from B to C * 4 highways from C to D = 24 different highway routes from city A to city D.
Therefore, there are 24 different highway routes from city A to city D using the fundamental counting principle.