Pascal's triangle, and the patterns hidden in it
One rule, applied repeatedly, produces most of elementary combinatorics.
Start with a 1. Each row begins and ends with 1, and every other entry is the sum of the two directly above it. That is the whole construction.
Every interior number is the sum of the two above it: 6 is 3 + 3, 15 is 5 + 10. One rule, applied over and over, and the structure that comes out of it is far richer than the rule suggests.
It is the combinations
Row n, position k gives the number of ways to choose k things from n. Row 5 reads 1, 5, 10, 10, 5, 1 - so from 5 objects there is 1 way to choose none, 5 ways to choose one, 10 ways to choose two, and so on.
The addition rule has a plain meaning once you see it. To choose 2 from 5, either you include the fifth object - then you need 1 more from the remaining 4 - or you do not, and you need all 2 from the remaining 4. So C(5,2) = C(4,1) + C(4,2), which is exactly the two numbers sitting above it.
That is a proof, and it came out of noticing what the rule means rather than manipulating formulas.
It is the binomial expansion
Multiply out (a + b) to various powers and the coefficients are the rows:
- (a+b)^1 = a + b -> 1, 1
- (a+b)^2 = a^2 + 2ab + b^2 -> 1, 2, 1
- (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 -> 1, 3, 3, 1
Which is not a coincidence. Expanding (a+b)^3 means choosing a or b from each of three brackets, and the coefficient of a^2b counts how many ways to pick b exactly once from three - a combination.
Other things living inside it
- Row sums are powers of two. 1, 2, 4, 8, 16, 32. Choosing any subset of n things gives 2^n options, and the row counts them by size.
- The second diagonal is the counting numbers 1, 2, 3, 4, 5.
- The third diagonal is the triangular numbers 1, 3, 6, 10, 15 - the running totals of 1, 2, 3, 4.
- Shallow diagonals sum to Fibonacci. Sum the diagonals at a shallow angle and you get 1, 1, 2, 3, 5, 8.
- Colour the odd entries and the Sierpinski triangle appears - a fractal, from a rule about adding neighbours.
It was not Pascal's. Versions appear in Chinese mathematics as Yang Hui's triangle from the 13th century, in Persian work by al-Karaji earlier still, and in Indian texts on poetic metre before that. Pascal wrote it up in 1654 and Europe attached his name. Worth knowing, because 'who discovered it' and 'whose name it carries' are different questions more often than you would expect.
A use you can check
Flip a coin 6 times. How many ways can you get exactly 2 heads? Row 6 is 1, 6, 15, 20, 15, 6, 1, so the answer is 15. Total outcomes are 2^6 = 64, so the probability is 15/64, about 23%.
And the shape of that row - small at the ends, large in the middle - is why extreme results are rare. Getting 3 heads out of 6 has 20 ways; getting 0 has one. That bulge in the middle, drawn for large n, is the bell curve, which is where statistics begins.