Greatest common divisor and least common multiple
Find the largest integer dividing both numbers and the smallest positive multiple shared by them. These answer different grouping and cycle questions.
MAKE THE FORMULA CLEAR
Combinations and permutations
Count selections when order does or does not matter. Objects must be distinct and cannot be selected more than once.
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Combinations: order does not matter C(n,r)
10ways
Selecting three people uses combinations. Assigning first, second and third place uses permutations because order matters.
Permutations: order matters P(n,r)60 ways
Worked steps
- Combinations C(5,3) = 5! ÷ (3! × 2!) = 10
- Permutations P(5,3) = 5! ÷ 2! = 60
- Permutations count different orders separately; combinations count the same selection only once.
Enter integers from 0 to 100 with r no greater than n. Objects are distinct and cannot be selected twice. Combinations ignore order; permutations count order. Choosing zero objects has one empty selection. Results are exact integers. Repetition, circular arrangements, identical objects and other constraints are unsupported.
Method and scope
Combinations use n! / [r! * (n-r)!], with the entire product r! * (n-r)! in the denominator. Permutations use n! / (n-r)!; a factorial multiplies the positive integers up to its argument, with 0! defined as one.
Selecting zero objects gives one empty selection. Exact integer arithmetic preserves long results rather than approximating them as floating-point numbers.
Enter integers from 0 to 100 with r no greater than n. Repetition, circular arrangements, identical objects and extra restrictions are excluded; a count alone is not a probability.
References:OpenStax: permutations and factorials、OpenStax: combinations and the role of order
What this helps you do
Count selections when order does or does not matter. Objects must be distinct and cannot be selected more than once.
What to enter
Start with number of distinct objects n, then fill the other relevant fields in the units shown. The field hints explain the required format and examples.
How the calculation works
Combinations use n! / [r! * (n-r)!], with the entire product r! * (n-r)! in the denominator. Permutations use n! / (n-r)!; a factorial multiplies the positive integers up to its argument, with 0! defined as one.
Worked example
Choosing three people from five gives ten unordered groups. Assigning three different roles from those five gives 60 ordered selections.
Understanding the answer
Selecting zero objects gives one empty selection. Exact integer arithmetic preserves long results rather than approximating them as floating-point numbers.
Scope and limitations
Enter integers from 0 to 100 with r no greater than n. Repetition, circular arrangements, identical objects and extra restrictions are excluded; a count alone is not a probability.