From basic counting to diamond-grid path sums — master the building blocks step by step. 20 problems, fully explained.
Choosing \(k\) items from \(n\) without caring about order.
Going from \((0,0)\) to \((m,n)\) moving only right or up: total paths = \(\binom{m+n}{m}\).
Sum over all paths = (contribution per edge) × (# paths through that edge). Swap the order of summation!
Every interior number in Pascal's Triangle equals the two above it.
In a diamond grid the area to the right of a path = the number of unit cells to the right of that path.
Count the same thing two ways, set them equal. Powerful trick for sum problems.