## Permutations and Combinations

### 3.3 Pascal’s Triangle

Lesson materials located below the video overview.

Turn the grid of numbers forty-five degrees to make a triangle of numbers:

The grid presented this way made famous by French mathematician Blaise Pascal (1623-1662) for his work in probability theory.

Each row of this triangle is a diagonal of the original grid and each entry in the triangle counts paths. (If you like, we can make this explicit by counting downward moving paths in a honeycomb.)

As before,

EACH INTERIOR ENTRY OF PASCAL’S TRIANGLE IS THE SUM OF THE

TWO ENTRIES ABOVE IT

We also have formulas for the individual entries of Pascalâ€™s triangle. The entry that was in the $$a$$th row and $$b$$th column in the original grid (starting our counts with zero-th rows and zero-th columns) is now in the $$(a+b)$$th row of Pascalâ€™s triangle, $$a$$ places in from the left and $$b$$ places in from the right along that row.

Regard the top row (consisting of the single 1) as the zero-th row of the triangle. The formula for the entry on row $$n$$, $$a$$ places in from the left and $$b$$ places in from the right is:

$$\dfrac{n}{a!b!}$$.

(And $$n$$ does equal $$a+b$$.)

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