Question

Mack the bug starts at (0,0) at noon and each minute moves one unit right or one unit up. He is trying to get to the point (5,7) . However, at (2,3) there is a spider that will eat him if he goes through that point. In how many ways can Mack reach (5,7) ?

Answer 1

let's look at it like this...

Mack has to move 5 to the right and 7 up to get to the point (5, 7), right?

Let's see if we can find how many total waysthere are to get to (5, 7) and then we'll see how many ways there are to get to (2, 3) and then subtract the two.

Answer 2

you there?

Answer 3

yes

Answer 4

to make it a little easier, let's look at how many way's there are to get to the point (2, 3)

Answer 5

let let r = a move to the right and u = a move up.

then we could get to (2, 3) in any of the following ways...

(r,r,u,u,u), (r,u,r,u,u), (r,u,u,r,u), (r,u,u,u,r),
(u,r,r,u,u), (u,r,u,r,u), (u,r,u,u,r),
(u,u,r,r,u), (u,u,r,u,r),
(u,u,r,r,r)

=10

but let's see if we can model using combinations because that will make our life easier to compute the number of ways to get to (5, 7)

Answer 6

really, there are 5 spots, 2 of which have to be to the right (or r).

\[\left(\begin{matrix}5 \\ 2\end{matrix}\right)=\frac{ 5! }{ 2!\left( 5-2 \right)! }=\frac{ 5! }{ 2!\,3! }=10\]

Answer 7

so can you compute how many ways there are to get to (5, 7)?

Answer 8

how many moves (spots) will Mack need to get to the point (5, 7)?

Answer 9

remember, Mack needed 5 moves to get to (2, 3)... a total of 2 to the right and 3 up, giving 2+3 = 5 total moves

Answer 10

oops, I made a little mistake... we'll have to subtract the number of ways to get to (5, 7) going through (2, 3) from the total.

Once we're at (2, 3) we have to move 3 to the right and 4 up. That means there are \[\left(\begin{matrix}7 \\ 3\end{matrix}\right)=35\]
ways of getting to (5, 7) from (2, 3). So well have to subtract 10*35 = 350 from our total number of ways of getting from (0, 0) to (5, 7).

Answer 11

Nevermind. Didn't see the restriction on possible actions, so deleted my question.

Answer 12

you there still?

Answer 13

make sense to you?

Answer 14

subtract (5 choose 2) * (4 choose 3) from it?

Answer 15

from (12 C 7) {=(12 C 5)}

Answer 16

Err, you're right. I forgot to include all of the steps in the part where we go from (2,3) to destination.

Answer 17

wait...so the answer is 350-35???

Answer 18

no, there are\[\left(\begin{matrix}12 \\ 7\end{matrix}\right)=792\]ways to get from (0, 0) to (5, 7) going only up or right.

there are 10*35 = 350 ways to get to (5, 7) from (0, 0) that go through the point (2, 3).

so Mack can go 792-350 = 442 different ways safely

Answer 19

ok. I get it. thanks

Answer 20

you're welcome

Answer 21

How many ways are there to put 6 balls in 3 boxes if three balls are indistinguishably white, three are indistinguishably black, and the boxes are distinguishable?

Answer 22

can you help with the problem above??

Answer 23

do as a new post... others may want to contribute. also, i'm leaving in a minute

Answer 24

how do you make a new post??

Answer 25

copy what you wrote and paste it in the box on the upper left that reads "Ask a question..."

Answer 26

good luck, gotta run