Total Population or Pool Size (n)Count (Whole Non-Negative Integer)
The total number of available items, choices, or distinct members in a collection from which you select or organize. In everyday selections without repeats (such as picking teammates or drawing lottery balls), your selection size r cannot exceed this total pool (r ≤ n). When items can be chosen multiple times (such as rolling a die or choosing frozen yogurt toppings), n represents the variety of distinct options available.
Default: n = 10 (Customizable from 0 to 10,000)
Sample Subset Size (r or k)Cardinality / Group Size (0 ≤ r ≤ n)
The exact number of slots to fill or items to choose from the pool. When the sequence matters (such as awarding 1st, 2nd, and 3rd place running medals), r represents the positions in a permutation P(n, r). When group composition is all that counts (such as picking a 3-person volunteer squad), r represents the group size in a combination C(n, r). Setting r = 0 always equals 1, representing the single way to leave all items unselected.
Default: r = 3 (Customizable from 0 to n)
Order Sensitivity & Replacement Flag (isOrdered, withReplacement)Arrangement Logic & Replacement Mode
Establishes counting rules across two practical questions: First, does order matter? In permutations, sequence counts (finishing 1st then 2nd is distinct from 2nd then 1st). In combinations, order is irrelevant (a fruit bowl of apples and bananas is identical to bananas and apples). Second, can choices be repeated? Without replacement means each item can be picked once; with replacement means items return to the pool for reuse.
Default: Order: Irrelevant (Combinations) | Replacement: Disabled
Multinomial Frequency Partitions (n1, n2, ... nk)Subgroup Partition Counts (Sum = n)
Used when arranging a group where certain items are identical to one another, such as rearranging the letters in 'CHEESE' or sorting red and blue sports jerseys. The multinomial calculation n! / (n1! × n2! × ... × nk!) divides out the unnoticeable swaps between identical items, providing the exact number of distinct, visually unique arrangements.
Default: Uniform Distinct Items (n1 = n2 = ... = nk = 1)
Derangement Subfactorial Operator (!n)Subfactorial Integer Sequence
Calculates complete reshuffles where not a single person or item ends up in their starting position—the classic rule of a Secret Santa gift exchange or returning winter coats to the wrong owners. As the group size grows beyond 4 people, the probability that a completely random swap leaves nobody with their own item rapidly settles at approximately 1/e (~36.79%).
Default: !n = floor(n! / e + 0.5) for n ≥ 1
Keyspace Entropy & Security Complexity (H = log2(S))Entropy Measure in Bits
Measures the total unpredictability and difficulty of guessing a sequence in binary bits (H = log2(Total Possibilities)). In personal digital safety, higher bit entropy means exponentially stronger defense against guessing attacks: a 6-digit numeric passcode (10^6) provides only ~19.9 bits of security (cracked in fractions of a second), whereas a 12-character passphrase chosen from letters, numbers, and symbols delivers ~78.6 bits of robust protection.
Default: H = log2(Total Outcomes) Bits
