# Pascal's Triangle

## 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

## 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
Homework Helper
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:
Code:
   B
I I
N N N
O O O O
M M M
I I
A
L L

HallsofIvy
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.