Casey walking from house to bus stop (WT, p. 62)

[Guest]

From p. 62 of the WT guide (combinatorics)
"Every morning, Casey walks from her house to the bus stop, as shown to the right. She always travels exactly nine blocks from her house to the bus, but she varies the route she takes every day. (One sample route is shown.) How many days can Casey walk from her house to the bus stop without repeating the same route?"

I'm confused on this one. The solution says an anagram of LLLLDDDDD (left 4, down 5), but this suggests that order doesn't matter. However, doesn't order matter in this case, since there's a different in going left first or down first?

Thanks.

[RonPurewal]

From p. 62 of the WT guide (combinatorics)
"Every morning, Casey walks from her house to the bus stop, as shown to the right. She always travels exactly nine blocks from her house to the bus, but she varies the route she takes every day. (One sample route is shown.) How many days can Casey walk from her house to the bus stop without repeating the same route?"

I'm confused on this one. The solution says an anagram of LLLLDDDDD (left 4, down 5), but this suggests that order doesn't matter. However, doesn't order matter in this case, since there's a different in going left first or down first?

Thanks.

you have just provided a nice public service for the other readers of this forum: you've illustrated the fact that 'order matters' isn't as simple of a concept as it might at first seem.

here's what might be a better way to think of the 'order matters' idea: instead of saying 'order doesn't matter', think of it as 'you can interchange them without changing the situation'. similarly, 'order matters' can be reinterpreted as 'interchanging them affects the situation'. in this case, if you permute the 'lefts', then you still most definitely have the same route; likewise if you permute the 'rights'.

put another way: if your left turns are blocks 2, 4, 6, 9, and your right turns are blocks 1, 3, 5, 7, 8, then that's the same situation as turning left at blocks 4, 9, 2, 6 and turning right at blocks 5, 3, 8, 7, 1. that's what we mean when we say order doesn't matter.

the fact that you must walk the blocks in a certain order is immaterial. that may be difficult to see, so try the rephrasing suggested above.

--

as final proof, consider a path that consists of nine straight blocks left - a path that obviously admits only one possibility.
under the reasoning used in the problem, this is the # of anagrams of LLLLLLLLL, which is 1.
with your reasoning ('order matters'), you'd be arranging nine different letters, for the ludicrous result that there are 9! = 362,880 different ways to walk a path consisting of nine straight blocks to the left.

good times!

-- ron

[Guest]

From p. 62 of the WT guide (combinatorics)
"Every morning, Casey walks from her house to the bus stop, as shown to the right. She always travels exactly nine blocks from her house to the bus, but she varies the route she takes every day. (One sample route is shown.) How many days can Casey walk from her house to the bus stop without repeating the same route?"

I'm confused on this one. The solution says an anagram of LLLLDDDDD (left 4, down 5), but this suggests that order doesn't matter. However, doesn't order matter in this case, since there's a different in going left first or down first?

Thanks.

you have just provided a nice public service for the other readers of this forum: you've illustrated the fact that 'order matters' isn't as simple of a concept as it might at first seem.

here's what might be a better way to think of the 'order matters' idea: instead of saying 'order doesn't matter', think of it as 'you can interchange them without changing the situation'. similarly, 'order matters' can be reinterpreted as 'interchanging them affects the situation'. in this case, if you permute the 'lefts', then you still most definitely have the same route; likewise if you permute the 'rights'.

put another way: if your left turns are blocks 2, 4, 6, 9, and your right turns are blocks 1, 3, 5, 7, 8, then that's the same situation as turning left at blocks 4, 9, 2, 6 and turning right at blocks 5, 3, 8, 7, 1. that's what we mean when we say order doesn't matter.

the fact that you must walk the blocks in a certain order is immaterial. that may be difficult to see, so try the rephrasing suggested above.

--

as final proof, consider a path that consists of nine straight blocks left - a path that obviously admits only one possibility.
under the reasoning used in the problem, this is the # of anagrams of LLLLLLLLL, which is 1.
with your reasoning ('order matters'), you'd be arranging nine different letters, for the ludicrous result that there are 9! = 362,880 different ways to walk a path consisting of nine straight blocks to the left.

good times!

-- ron

thanks ron. however, i'm still trying to think in terms of walking the blocks in a certain order. i understand how the anagram model works for other problems (e.g. how many different teams of 4 people can be chosen from a total of 7 people in the room, for which the anagram would be YYYYNNN), but here it matters that which blocks be walked in a certain order. in other words, LLLLDDDDD means that i take 4 lefts and 5 downs regardless of sequence, but each of those permuations still changes my "situation" - which in this case is each path itself, right?

[StaceyKoprince]

Don't think of the anagram method as "she goes left for the first block because I've got an L in the first square, and then she goes left for the second block because I've got an L in the second square." Instead, what you're saying is "she goes left once at some point in the sequence because I've got an L in the first square but that doesn't necessarily mean she goes left FIRST." Etc. The linear order in the anagram does not (necessarily) match the linear order of the blocks that Casey walks.

How many times will she have to go left? 4. How many times will she have to go down? 5. Can she do those actions for different blocks? Yes, she could go left first or down first, as long as she goes left 4 times and down 5 times in total.

Also, don't worry if you're just not "getting" this intuitively - most people don't "get" combinatorics. First, know that the anagram method has built in the idea that this set-up will calculate the # of unique ways in which she can go L4 and D5. Second, you're unlikely to see more than one combinatorics question on the test, so don't stress over this are too much. :)