The first few rows of Pascal's Triangle are as follows:
1 Row 0
1 1 Row 1
1 2 1 Row 2
1 3 3 1 Row 3
1 4 6 4 1 Row 4
1 5 10 10 5 1 Row 5
1 6 15 20 15 6 1 Row 6
1 7 21 35 35 21 7 1 Row 7
1 8 28 56 70 56 28 8 1 Row 8
. . .
This triangle is named after Blaise Pascal, who wrote Treatise on
the Arithmetical Triangle in 1653. While Pascal was not the first
person to discover this binomial triangle (the Persians and Chinese had
used it centuries ago, and even in Europe Tartaglia had mentioned this
triangle in his General Trattato), Pascal was the first to
discover many of the triangle's interesting properties. Each number in
Pascal's triangle is the sum of the (usually two) numbers directly above it.
Some of the interesting properties of Pascal's triangle are as follows:
- The sum of the numbers in row n is 2 n.
- The first diagonal is occupied by 1's.
- The second diagonal is occupied by the
natural numbers (1, 2, 3, 4, etc.).
- The third diagonal is occupied by
triangular numbers (1, 3, 6, etc.).
- The fourth diagonal contains tetrahedral numbers.
- The rth number in the nth
row is equal to nCr (where C is the combination
- Interesting patterns can be formed by
looking at the remainders after dividing
each number by a given number. There are some shortcuts that can be
employed here - one doesn't have to calculate all of the numbers in
a certain section of the triangle. For example, Lucas' theorem states
that, to determine whether the number in the kth
column of the
nth row of Pascal's triangle is even or odd,
convert n and k to base 2.
The number is even unless every binary digit of k is less than
or equal to the corresponding binary digit of n.
- Fibonacci numbers can be extracted from the triangle (see the
Fibonacci numbers page for more information).