# How many paths spell MICROWAVE

    M
I I
C C C
R R R R
O X O O O
W W W W
A A A
V V V
E E


Starting with the M on top and only moving one letter at a time down to the left or right, how many different paths from top to bottom spell MICROWAVE? However you cannot move to the "X". I understand this uses Pascal's triangle but I'm not sure where to go once I reach the "X" since, it needs to be skipped. I would really appreciate it if someone could help me with this problem.

    1
1 1
1 2 1
1 3 3 1
1 X 6 4 1
W W 10 5
A A A
V V V
E E


The work above is when I tried to solve this using Pascal's triangle, but as you can see above, I'm not sure which number to put after for "W".

• Are the final V and E lines really supposed to have 3 and 2 positions? Can you go from the middle A to any of the three Vs? Jul 8, 2021 at 22:25
• Yes they are and no you can't move to any of the Vs. It can only move down left or right one time
– user948479
Jul 8, 2021 at 22:27
• If you can't move to any of the Vs, there is surely no way to spell MICROWAVE which has a V as its penultimate letter. Jul 8, 2021 at 22:32
• @Troposphere What Mandy means is that you can only move from an A to an adjacent V. Jul 8, 2021 at 22:33
• Was it your intent that each V be to the right of an A in the line above it? Jul 8, 2021 at 22:34

## 2 Answers

Originally I had an overcomplicated way, but since you brought up Pascal's triangle I will continue from there.

You are close to the answer, the final step you need is to write $$X$$ as $$0$$ simply, as there is "no way" to get to that square and of course no way to get out. You can then continue the Pascal's triangle:

    1
1 1
1 2 1
1 3 3 1
1 0 6 4 1
1 6 10 5
7 16 15
23 31 15
54 46


(If you want, you can add a "sink" at the end that collects both E's that will have a value of 100, but that's not necessary).

Finally, the answer is $$54+46=100$$.

Place a 1 on the M.

The Is both become 1s

The line of Cs become 121 - just like Pascal’s triangle. That’s because the number of ways of getting to a particular letter is the sum of the two above.

The line of Rs become 1331

The line of Os would become 14641 but one of them is forced (by the X) to be 0.

That makes a line that is 10641

The line of Ws is 1 6 10 5

The line of As is 7 16 15

The line of Vs is 23 31 15

I’ll leave the last line to you!

The final answer is the sum of the last two Es.