Pascal's Triangle: 40 Paths to Spell BINOMIAL

[kerrwilk]

Homework Statement

How many different paths will spell the word BINOMIAL in the following arrangement(moving diagonally downwards to the left or right)?

...B
...I I
..N N N
O O O O
.M M M
...I I
...A
...L L

Homework Equations

The Attempt at a Solution

Starting from B using Pascal's Triangle

...B
..1 1
.1 2 1
1 3 3 1
.4 6 4
.10 10
...20
.20 20

There are 40 different paths. Is this correct? Thanks for your help.

Note: I only put dots to center the pattern. They have no meaning.

 

[eumyang]

How many different paths will spell the word BINOMIAL in the following arrangement(moving diagonally downwards to the left or right)?

...B
...I I
..N N N
O O O O
.M M M
...I I
...A
...L L

You can use the CODE tags to line up letters better, like this:

   B
  I I
 N N N
O O O O
 M M M
  I I
   A
  L L

 

[HallsofIvy]

It's not clear what you mean by that. You title this "Pascal's triangle" but you appear to have the number of letters in each row coming back to 1 at "A" and than back to 2 at "L".

If you really mean Pascal's triangle so that the number of letters in each row increases so that there are 8 "L"s in the final row, then at each step (at each letter) except the last, you have 2 choices which way to go. Since you are making 7 choices, you have $2^7$ possible paths.