There are 4 roads from town A to town B, and 3 roads from town B to town C. In how many ways could a person travel
(a) from A to C, passing through B?
(b) round trip from A to C and back, by way of B?
(c) round trip from A to C, by way of B without using any road twice?
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Are the following correct?
(a) 12
(b) 144
(c) 72
(a) from A to C, passing through B?
(b) round trip from A to C and back, by way of B?
(c) round trip from A to C, by way of B without using any road twice?
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Are the following correct?
(a) 12
(b) 144
(c) 72
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(a) Your answer to this one is correct. There are 4 ways to get to B from A, and there are 3 ways to get to C from B; therefore, by the Fundamental Counting Principle, there are 4*3 = 12 ways to get from A to C while passing through B.
(b) There are 12 ways to get from A to C through B, and there are 12 ways back from C to A through B, so there are 12*12 = 144 ways to get to A to C through B and back through B and you are correct.
(c) There are 12 ways to get from A to C through B. Now, since we cannot use any road twice, we have to pick a different road to return to B than the one we chose to get from A to B; this leaves 2 possible roads to get from C to B without using the same road we chose to get from A to B. In a similar fashion, there are only 3 ways to get from B to A without using the same road we chose to get from A to B. By the Fundamental Counting Principle, there are 12*2*3 = 72 ways to do this and you are correct here as well.
I hope this helps!
(b) There are 12 ways to get from A to C through B, and there are 12 ways back from C to A through B, so there are 12*12 = 144 ways to get to A to C through B and back through B and you are correct.
(c) There are 12 ways to get from A to C through B. Now, since we cannot use any road twice, we have to pick a different road to return to B than the one we chose to get from A to B; this leaves 2 possible roads to get from C to B without using the same road we chose to get from A to B. In a similar fashion, there are only 3 ways to get from B to A without using the same road we chose to get from A to B. By the Fundamental Counting Principle, there are 12*2*3 = 72 ways to do this and you are correct here as well.
I hope this helps!
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a) Yes, that is correct. By the Multiplication Principle, we have 3*4 = 12.
b) Yes, that is correct as well. Once we have 12 from A to C, from C to A, we have 12 as well. The route taken from A to C is independent of the route from C to A. So 12*12 = 144.
c) There are 12 initial ways to get from A to C. On the way back, we have to eliminate the use of 2 roads (the one from A to B and the one from B to C). So we multiply: 12*(4-1)(3-1) = 12*3*2 = 72.
Good job!
b) Yes, that is correct as well. Once we have 12 from A to C, from C to A, we have 12 as well. The route taken from A to C is independent of the route from C to A. So 12*12 = 144.
c) There are 12 initial ways to get from A to C. On the way back, we have to eliminate the use of 2 roads (the one from A to B and the one from B to C). So we multiply: 12*(4-1)(3-1) = 12*3*2 = 72.
Good job!