Decimal to Binary Converter

Binary Result:

Combinations Calculator

Calculate the number of ways to choose r items from n items (order doesn't matter)

Permutations Calculator

Calculate the number of ways to arrange r items from n items (order matters)

Key Differences:

Combinations: Order doesn't matter. Choosing ABC is the same as choosing CBA.

Permutations: Order matters. ABC and CBA are counted as different arrangements.

Example: Choosing 3 team members from 5 people uses combinations (order doesn't matter). Assigning president, vice-president, and secretary from 5 people uses permutations (order matters).

Three separate tools on one page

This page bundles a decimal-to-binary converter with two counting calculators (combinations and permutations) because all three rely on the same "break a number down step by step" teaching approach, even though the underlying math is unrelated.

Decimal to binary: worked example

Converting 42 uses repeated division by 2, keeping the remainder at each step: 42÷2=21 r0, 21÷2=10 r1, 10÷2=5 r0, 5÷2=2 r1, 2÷2=1 r0, 1÷2=0 r1. Reading the remainders from the last division back to the first gives 101010. Verify it: 1×32 + 0×16 + 1×8 + 0×4 + 1×2 + 0×1 = 32+8+2 = 42. The tool performs this same verification automatically and shows it under the step list.

Combinations: worked example

Combinations count selections where order doesn't matter — C(n,r) = n! / (r! × (n-r)!). Choosing 3 people from a group of 5 (C(5,3)): 5! / (3! × 2!) = 120 / (6 × 2) = 10. There are 10 distinct 3-person groups possible, and for n ≤ 10 with a small enough result the tool lists every one of them.

Permutations: worked example

Permutations count arrangements where order matters — P(n,r) = n! / (n-r)!. Assigning 3 distinct roles (say, 1st/2nd/3rd place) among the same 5 people (P(5,3)): 5! / 2! = 120 / 2 = 60. That's six times more than the combination count, because each group of 3 can be ordered 3! = 6 different ways — the exact ratio between P(5,3) and C(5,3).

Constraints

The binary converter only accepts non-negative integers. For combinations and permutations, r cannot exceed n, and both must be non-negative; the full listing of outcomes only renders when the counts are small enough to display without flooding the page (n≤10 and result≤100 for combinations, n≤8 and result≤100 for permutations).