In this city, all the streets that run North and South have lettered names (A,B,C, etc.) and all the streets that run East-West have numbered names (1st, 2nd, 3rd, etc.). As you drive East, the letters get later in the alphabet. As you drive south, the numbers get bigger. Mark lives at the corner of A and 1st. His girlfriend, Diane lives at the corner of E and 8th. Mark enjoys riding his bike to Diane's house. However, he likes to go a different way every day. If he only rides east and south, how many different routes can he take from his house to Diane's house?
I'm a little confused. I figure going East would look like..
_____A street.___________B st. _______C st._________D st.___________E st.
Then, going South it would be downward..
1st
-
2nd
-
3rd
-
4th
etc..
I'm guessing there is a possibilty he could take all the options. Such as he could go A street and 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th. Than for B street he can go B then 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th.. continuing this pattern until he went straight to e street and then down 1st, 2nd, etc.
Going from this way the answer would be 40 different routes. Is this the way to set up this problem? If not explain how you would solve it please.
Hint: Our cyclist wants to go a total of $4$ blocks eastward, and $7$ blocks southward, a total of $11$ blocks. His trip can be described by an $11$ letter "word" in the alphabet {E, S}, that has exactly $4$ E's. Any trip can be described by such a word, and any such word corresponds to a legal trip. Mark must choose where the $4$ E's will be.