The Basics
This is the number located at the very top of the triangle (Row 0).
1
The sum of any row n is equal to this base raised to the power of n
2
This formula, which starts (x+y)^n), uses the triangle’s rows to find coefficients.
Binomial Theorem
Although named after Blaise Pascal, the triangle was studied centuries earlier in this country by mathematician Yang Hui.
China
If you square every number in Row (n) and add them up, you get the middle number of Row (2n).
Sum of Squares
To find a value in the triangle, you add this many numbers from the row directly above it.
two
If you shade only the odd numbers in the triangle, you create this famous fractal.
Sierpiński Triangle
In "n choose k" notation, the "n" represents this part of the triangle.
Row
This was the primary century during which Blaise Pascal lived and published his "Treatise on the Arithmetical Triangle."
17th Century
This "sports-themed" identity says the sum of a diagonal starting from a 1 equals the number below and to the left of the last term.
Hockey Stick Identify
This is the specific value of the 3rd number in Row 4 (1, 4, 6, 4, 1)
6
Summing the numbers along shallow diagonals of the triangle produces this famous sequence.
Fibonacci sequence
To find the coefficients for (x+y)^2), you would look at this specific row.
Row 2 (1, 2, 1)
Before it was "Pascal’s," Persian mathematicians referred to it as the triangle of this famous poet and scholar.
Omar Khayyam
According to this rule, the alternating sum of any row (e.g., (1 - 4 + 6 - 4 + 1) always equals this value.
Zero
Because the left side is a mirror image of the right, the triangle is said to have this geometric property.
symmetry
The second diagonal (1, 2, 3, 4...) contains the counting numbers, but the third diagonal (1, 3, 6, 10...) contains these types of numbers.
Triangular Numbers
Pascal’s Triangle is often used to calculate these, which represent the number of ways to choose a subset from a larger set.
Combinations
Pascal originally developed the triangle to help solve problems related to this "vice" involving dice and coins.
Gambling.
This identity describes the relationship where any number in the triangle is the sum of the two numbers above it.
Every row in the triangle begins and ends with this integer.
1
If the first number after the "1" in a row is this type of number, then every other number in that row (excluding the 1s) is divisible by it.
Prime Number
This is the coefficient of the (xy^{3}) term in the expansion of (x+y)^4).
4
This is the name of the rule stating that the sum of a diagonal of any length equals the number below the last entry but not in that diagonal.
Hockey Stick Identity
If you treat each row as digits (carrying over when numbers are (>9), Row (n) represents this number raised to the (n)th power.
11