Editing Pascal's triangle (section)

=== Overall patterns and properties ===
[[File:Sierpinski Pascal triangle.svg|thumb|A level-4 approximation to a [[Sierpiński triangle]] obtained by shading the first 32 rows of a Pascal triangle white if the binomial coefficient is even and black if it is odd.]]
* The pattern obtained by coloring only the odd numbers in Pascal's triangle closely resembles the [[fractal]] known as the [[Sierpiński triangle]]. This resemblance becomes increasingly accurate as more rows are considered; in the limit, as the number of rows approaches infinity, the resulting pattern ''is'' the Sierpiński triangle, assuming a fixed perimeter. More generally, numbers could be colored differently according to whether or not they are multiples of 3, 4, etc.; this results in other similar patterns.
:As the proportion of black numbers tends to zero with increasing ''n'', a corollary is that the proportion of odd binomial coefficients tends to zero as ''n'' tends to infinity.<ref>Ian Stewart, "How to Cut a Cake", Oxford University Press, page 180</ref>
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| style="padding: 0; vertical-align: inherit; background-color: #d18b47;" | [[File:Chess x1d45.svg|26x26px|alt=b4 one|b4 one]]
| style="padding: 0; vertical-align: inherit; background-color: #ffce9e;" | [[File:Chess x1l45.svg|26x26px|alt=c4 one|c4 one]]
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| style="padding: 0; vertical-align: inherit; background-color: #d18b47;" | [[File:Chess x3d45.svg|26x26px|alt=c3 three|c3 three]]
| style="padding: 0; vertical-align: inherit; background-color: #ffce9e;" | [[File:Chess x4l45.svg|26x26px|alt=d3 four|d3 four]]
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Pascal's triangle overlaid on a grid gives the number of distinct paths to each square, assuming only rightward and downward steps to an adjacent square are considered.
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* In a triangular portion of a grid (as in the images below), the number of shortest grid paths from a given node to the top node of the triangle is the corresponding entry in Pascal's triangle. On a [[Plinko]] game board shaped like a triangle, this distribution should give the probabilities of winning the various prizes.

[[Image:Pascal's Triangle 4 paths.svg|center|400px]]

* If the rows of Pascal's triangle are left-justified, the diagonal bands (colour-coded below) sum to the [[Fibonacci number]]s.
::{| style="align:center;"
|- align=center
|bgcolor=red|1
|- align=center
| style="background:orange;"|1
| style="background:yellow;"|1
|- align=center
| style="background:yellow;"|1
|bgcolor=lime|2
|bgcolor=aqua|1
|- align=center
|bgcolor=lime|1
|bgcolor=aqua|3
| style="background:violet;"|3
|bgcolor=red|1
|- align=center
|bgcolor=aqua|1
| style="background:violet;"|4
|bgcolor=red|6
| style="background:orange;"|4
| style="background:yellow;"|1
|- align=center
| style="background:violet;"|1
|bgcolor=red|5
| style="background:orange;"|10
| style="background:yellow;"|10
|bgcolor=lime|5
|bgcolor=aqua|1
|- align=center
|bgcolor=red|1
| style="background:orange;"|6
| style="background:yellow;"|15
|bgcolor=lime|20
|bgcolor=aqua|15
| style="background:violet;"|6
|bgcolor=red|1
|- align=center
| style="background:orange; width:40px;"|1
| style="background:yellow; width:40px;"|7
| style="background:lime; width:40px;"|21
| style="background:aqua; width:40px;"|35
| style="background:violet; width:40px;"|35
| style="background:red; width:40px;"|21
| style="background:orange; width:40px;"|7
| style="background:yellow; width:40px;"|1
|}