Count the ways to pick r items from n — ordered or not, with repetition or not. Every figure is computed exactly with big integers, so the digits you see are the digits, and the classic P(n,r) and C(n,r) sit alongside the two repetition cases people usually mean instead.
Try
The four ways to countpicking 3 from 10
No repetition
With repetition
Order matters
P(n,r) = n! / (n−r)!
720
Podium finishes — gold, silver, bronze from the same field
nʳ
1 000
A PIN — each position independently picks any of the n symbols
Order does not matter
C(n,r) = n! / (r! (n−r)!)
120
A lottery draw, or a hand of cards
C(n+r−1, r)
220
Scoops of ice cream — you may take the same flavour twice
The two shaded cells are what “permutations” and “combinations” usually mean.
How they relate
P(10, 3) = C(10, 3) × 3! = 120 × 6 = 720
Each unordered selection can be arranged 3! = 6 ways, which is exactly the factor between the two.
n!
3 628 800
7 digits
r!
6
1 digits
(n−r)!
5 040
4 digits
Where r sitsC(10, k) for every k — your r is highlighted
0
1
2
3
4
5
6
7
8
9
10
The curve is symmetric because C(n, k) = C(n, n−k): picking 3 to keep is the same as picking 7 to leave.