Poker Hands, Pixel Rotations and the Speed of Light in One Line

Three new math calculators: combinations and permutations for counting problems, matrix operations for linear algebra, and scientific notation converter with SI prefixes.

Patryk Matyjasik · 18 June 2026

Every time you shuffle a deck of cards, you create an arrangement that has almost certainly never existed before. Every time your phone rotates a photo, a tiny matrix multiplication runs behind the screen. Every time a physics textbook writes the mass of an electron, it compresses 31 zeros into five characters. Three everyday events, three branches of mathematics, three new calculators on LiczGrupa.

2,598,960 poker hands and one lottery ticket

A standard poker hand is 5 cards drawn from a deck of 52. The order you receive them does not matter - only which five cards you hold. That makes it a combinations problem.

C(52, 5) = 52! / (5! x 47!) = 2,598,960

That is roughly 2.6 million distinct hands. Among them, exactly 4 are royal flushes - one per suit. The probability of being dealt a royal flush is 4 / 2,598,960, about 0.000154 percent.

Now change the question. A lottery draws 6 numbers from 49. Same formula, different inputs:

C(49, 6) = 13,983,816

Your chance of matching all six: roughly 1 in 14 million. The calculator shows both results in under a second. It also shows the permutation count P(n, r) and the variation count n^r, so you can compare what happens when order matters or when repetition is allowed.

Combinations and Permutations Calculator showing poker hand calculation with n=52, r=5

The Combinations & Permutations Calculator takes two inputs - n (total elements) and r (how many you choose) - and returns three values at once. Combinations ignore order. Permutations count order. Variations allow repetition. One click, three answers, zero ambiguity about which formula to use.

Rotating a pixel is multiplying a matrix

When you rotate an image 90 degrees on your screen, every pixel moves to a new position. The math behind that move is a 2x2 matrix multiplication:

``

1 0xy=x
``

The rotation matrix [[0, -1], [1, 0]] maps every point (x, y) to (-y, x). That is a 90-degree counterclockwise rotation. Its determinant is 0 x 0 - (-1) x 1 = 1, which confirms it preserves area (no stretching, no squishing).

Matrices show up everywhere once you start looking: computer graphics, machine learning weight layers, solving systems of equations, Markov chains in statistics. The core operations are always the same: addition, multiplication, transpose, determinant, inverse.

Matrix Calculator showing determinant of a 2x2 matrix with result det(A) = 8

The Matrix Calculator supports seven operations on 2x2 and 3x3 matrices. Pick the operation from a dropdown, fill in the elements, and the result appears with an explanation of the formula used. Need the inverse of [[3, 7], [1, 5]]? The calculator finds det = 8, then computes A^-1 = (1/8) x [[5, -7], [-1, 3]]. It also catches singular matrices (det = 0) and tells you when the inverse does not exist.

299,792,458 meters per second in five characters

The speed of light is 299,792,458 m/s. Writing that number every time is impractical. Scientific notation compresses it:

2.998 x 10^8

Four significant figures, one power of ten, done. The same notation handles the other extreme: the mass of a proton is 0.00000000000000000000000000016726 kg, which becomes 1.673 x 10^-27 kg. Without scientific notation, physics papers would be unreadable.

The calculator goes further. It shows three formats at once:

  • Scientific notation: 2.998 x 10^8
  • Engineering notation: 299.8 x 10^6 (exponent divisible by 3, aligned with SI prefixes)
  • E notation: 2.998E8 (the format used in code and spreadsheets)

It also identifies the SI prefix. For 299,792,458 the engineering exponent is 6, which corresponds to M (mega) - roughly 300 megameters per second.

Scientific Notation Calculator converting 299792458 to 2.998 x 10^8

The Scientific Notation Calculator works in both directions. Type a number and get the scientific form. Or type a mantissa and exponent and get the full number back. Precision is adjustable from 2 to 10 significant figures.

Three tools, one pattern

Combinations count possibilities without caring about order. Matrices transform coordinates by multiplying rows and columns. Scientific notation rescales numbers by powers of ten. Different operations, but the same underlying idea: take something complex and express it in a compact, standard form that anyone can read.

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