PERMUTATIONA in Apple Numbers: Ordered Selections With Repeats

PERMUTATIONA: count ordered sequences where choices may repeat. Order matters and a choice can repeat in multiple positions. Example: =PERMUTATIONA(3,2) has the expected mathematical result 9. Count possible access-code patterns, sequences and arrangements with reuse of available symbols. [A1, M1]

=PERMUTATIONA(num, num-chosen)

• num (required, number): Number of available possible items for each position; must be 0 or greater. Apple says the fractional part of a numeric argument is ignored.
• num-chosen (required, number): Number of ordered positions; must be 0 or greater. Apple says the fractional part of a numeric argument is ignored.

Both arguments are required numeric values; no default value is documented for either. Argument order is significant: first supply the size of the available collection, then the number of items/positions chosen. The function returns a count of outcomes, not a list of the actual outcomes. Apple explicitly ignores fractional parts. A positive number of positions with zero available choices is an edge case not exercised in Numbers here. [A1]

How to interpret the count mathematically

For whole numbers, the calculation is n^k for whole-number inputs n > 0 and k ≥ 0. This expression is an independent combinatorial identity and provides a convenient check of the expected result. It does not establish undocumented app handling of errors, overflows, large arguments or numeric coercion. [A1, M1]

The English function identifier and syntax above were checked in Apple Formulas and Functions Help 15.4 on 2026-10-12T03:42:00.411513Z. The version 15.4 labels the help edition consulted, not a confirmed minimum release of the Numbers application. Apple’s general guide discusses Numbers on supported Apple devices and web platforms; these four individual articles do not independently certify every historical macOS/iOS/iPadOS/iCloud edition. Translated function names or alternate spellings were not established; none are claimed as aliases. [A1, A2]

Apple’s documented examples use , as the function argument separator. If your decimal separator is a comma, Apple says to use a semicolon between arguments when typing formulas; for example =PERMUTATIONA(3;2) instead of =PERMUTATIONA(3,2). The formula editor can insert the correct locale punctuation. Function names can be typed in upper or lower case, and parentheses enclose the arguments. [A2]

Cell references follow Numbers conventions: B2 is a same-table cell, and B2:B6 denotes a range. For a cell in another table, Apple documents a qualified reference such as Table 2::B2; for another sheet, Sheet 2::Table 2::B2 is an example. If table names contain spaces, select cells in the formula editor to insert correctly formatted references; do not substitute Excel’s Sheet1!B2 syntax. Below, all formulas use two single numeric cells, not a range argument. [A3]

Verification status: Official Apple documentation checked. Example outcomes independently calculated using Python’s standard-library arithmetic/combinatorics. Not executed in Apple Numbers. Displayed answers are expected, not observed native results. No genuine Apple Numbers screenshots were taken; there are no mock app images in this guide.

Create a Numbers table named SelectionCounts. Place Available in B1, Chosen in C1 and Result in D1. Enter these whole-number inputs into rows 2–6, enter =PERMUTATIONA(B2,C2) in D2, and fill the formula down through D6. The expected numbers are independently computed, not observations in Numbers. [A1, A3, M1]

Row: 2

B: Available
3

C: Chosen
2

D formula
=PERMUTATIONA(B2,C2)

Expected D
9

Row: 3

B: Available
5

C: Chosen
3

D formula
=PERMUTATIONA(B3,C3)

Expected D
125

Row: 4

B: Available
4

C: Chosen
0

D formula
=PERMUTATIONA(B4,C4)

Expected D
1

Row: 5

B: Available
8

C: Chosen
2

D formula
=PERMUTATIONA(B5,C5)

Expected D
64

Row: 6

B: Available
10

C: Chosen
4

D formula
=PERMUTATIONA(B6,C6)

Expected D
10,000

A reader can also enter these standalone expressions in empty cells to avoid issues caused by table naming or references:

For the first row, use your own labels to enumerate the outcomes and verify the result by hand before trusting it for a larger dataset. Changing B or C changes the count, but the formula does not generate every actual selection. [M1]

Application 1 — ordered code patterns

Build a two-symbol ordered code using three permitted symbols, allowing repeats such as AA. The number of codes is 9. Create a fresh table and place 3 in B2 and 2 in C2; =PERMUTATIONA(B2,C2) has expected result 9. This result assumes positions are labelled, all choices are available at each position and repeated choices are allowed. [A1, M1]

Application 2 — calculate fixed-length code space

Build a four-symbol ordered code from ten possible digits, assuming leading zeros are allowed and digits may repeat. There are 10,000 codes; this is a counting exercise, not a recommendation for security-sensitive PIN design. Enter 10 in B3 and 4 in C3. =PERMUTATIONA(B3,C3) has expected result 10,000. This is a count of logically possible outcomes under the stated assumptions; real-world constraints such as availability, limited stock, banned sequences or people declining assignments reduce the actually available outcome space. [M1]

A tiny exhaustive example

Two positions, three choices each: AA, AB, AC, BA, BB, BC, CA, CB, CC. All nine are distinct ordered sequences, including AA, BB and CC with repetition. Thus =PERMUTATIONA(3,2) has expected result 9, compared with 6 for =PERMUT(3,2) (no repeats). [A1, M1]

Quick independent cross-check with exponents

Apple documents the identity PERMUTATIONA(n,k)=n^k. Consequently, =PERMUTATIONA(5,3) has expected result 125, the same as the mathematical value of 5^3. Apple Numbers’ POWER function or an exponent calculation can serve as a separate mathematical cross-check when manually validating a sheet. This guide has not executed either formula in Numbers. [A1, M1]

When should you use which related function?

Counting question: Choose an unordered unique group

Order significant?
No

Reuse the same item?
No

Function
COMBIN

Counting question: Choose an unordered group with repeats

Order significant?
No

Reuse the same item?
Yes

Function
COMBINA

Counting question: Assign ordered positions with no repeats

Order significant?
Yes

Reuse the same item?
No

Function
PERMUT

Counting question: Build ordered sequences with repeats

Order significant?
Yes

Reuse the same item?
Yes

Function
PERMUTATIONA

The same inputs (3 available items and 2 chosen) produce 3 (COMBIN), 6 (COMBINA), 6 (PERMUT) and 9 (PERMUTATIONA). The two middle results happen to match at 3/2; that does not make the functions interchangeable. Choose based on order and replacement, not solely because a trial output looks plausible. [A1, M1]

Zero items selected, large results and numeric precision

For four available options and zero chosen positions, the expected count is 1: the one empty choice/sequence. Apple’s article for PERMUTATIONA explicitly demonstrates the corresponding zero-group example. The exercise =PERMUTATIONA(4,0) has mathematically expected result 1 and has not been executed in Numbers. [A1, M1]

Combinatorial results can grow extremely quickly with the input size. A calculation may exceed practical displayed precision or spreadsheet numeric limits. For a business decision, validate that your input model and assumptions are reasonable; avoid interpreting a rounded, scientific-notation display as proof of an exact integer stored by the application. No native overflow threshold is asserted here.

Decimal arguments: documented truncation

Apple explicitly drops decimal portions of both numeric arguments for PERMUTATIONA. For =PERMUTATIONA(3.9,2.2), the truncated arguments are 3 and 2; the expected answer is 9. This expectation follows the documented rule and an independent mathematical calculation, not an observation in the app. If your input values are measurements or statistical estimates, decide explicitly whether truncating them is appropriate instead of silently losing precision. [A1, M1]

For errors, do not infer an exact Numbers error message from this guide. Apple’s individual function entry documents argument constraints but not a comprehensive native error-code matrix. This matters when migrating a workbook from Excel, where a visually similar function may return differently labelled error text. [A1, X1]

Microsoft’s official Excel documentation also lists the function PERMUTATIONA and describes the corresponding ordered selections with repetition for current documented Excel editions. The common positive whole-number examples above have the same mathematical interpretation. The argument names differ between Apple’s documentation and Excel’s; cross-file formula conversion, error strings, rounding/coercion details, date/platform availability, separators and references must be checked independently rather than assumed identical. [X1, A1]

Google Sheets, LibreOffice and other apps were not natively tested for this guide. Do not infer that this exact function identifier or its input boundaries exist unchanged in every program. In particular, use Numbers’ ::-qualified cross-table references rather than pasting an Excel Sheet!Cell reference unchanged. [A3]

• [A1] Apple, PERMUTATIONA. Formulas and Functions Help, edition 15.4, accessed 2026-10-12T03:42:00.411513Z. Official function article. Supports: exact syntax and argument order, documented numeric constraints and integer coercion, function meaning and explicit Apple examples. The supporting calculation records maps individual claims to article wording and independently computed vectors.
• [A2] Apple, Functions overview, edition 15.4, accessed 2026-10-12T03:42:00.411513Z. Official function syntax and locale rules. Supports: required vs optional arguments, capitalization, parentheses, and semicolon delimiter for decimal-comma locales.
• [A3] Apple, Refer to cells in formulas, edition 15.4, accessed 2026-10-12T03:42:00.411513Z. Official Numbers reference syntax. Supports: cell and range references, double-colon table/sheet qualifications and copy/absolute reference rules.
• [A4] Apple, List of functions by category, edition 15.4, accessed 2026-10-12T03:42:00.411513Z. Official catalogue. Supports canonical English spelling and category; does not certify minimum historical device/release support.
• [X1] Microsoft Support, PERMUTATIONA function for Excel, accessed 2026-10-12T03:42:00.411513Z. Official Excel reference. Supports only the Excel-specific comparison and existence claim, not a Numbers error-code claim.
• [M1] Independent mathematical check with Python math.comb / math.perm / integer exponent operations and a manual enumeration for the small case. Input/output vectors and calculation method are preserved in the evidence JSON. This is not native Apple Numbers execution.

• FACT — continue with a related Apple Numbers calculation.
• FACTDOUBLE — continue with a related Apple Numbers calculation.
• COMBIN — continue with a related Apple Numbers calculation.
• COMBINA — continue with a related Apple Numbers calculation.
• PERMUT — continue with a related Apple Numbers calculation.

For a software-specific comparison: LibreOffice Calc PERMUTATIONA. A shared name does not establish identical argument, error or version behaviour.

The linked official function documentation was reviewed on October 12, 2026. Apple’s Formulas and Functions Help 15.4 identifies the documentation edition, not a minimum Numbers release or a guarantee of historical platform parity. Selected worked examples were checked using independent mathematical or text-processing models; results are expected values, not observations from the spreadsheet application. Native Apple Numbers execution was not performed. Undocumented coercion, exact error messages and platform-specific behaviour remain unverified.

View reproducible example inputs and independent check results. This record is not a native spreadsheet test log.