# 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.

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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.