PERMUT

Microsoft publishes =PERMUT(A2,A3) = 970200 for 100 objects taken 3 at a time. DERIVATION in exact integer arithmetic from the documented equation n!/(n-k)!: 100 x 99 x 98 = 970200 exactly. No factorial of 100 is ever formed -- the ratio telescopes to the product of the top k terms, which is both the honest reading of the formula and the only way to evaluate it without overflow. Microsoft's page was re-read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/permut-function (the /en-us/office/<name>-function-<guid> path was serving Microsoft's 'Sorry, the page you're looking for can't be found' body throughout this batch, and the working path returns an 87 KB stub about half the time, so pages were fetched with retries until the payload exceeded 150 KB).

Microsoft publishes =PERMUT(3,2) = 6, "Permutations possible for a group of 3 objects where 2 are chosen". 3!/1! = 3 x 2 = 6, and the six orderings can be written out by hand, so this case is verifiable without arithmetic at all.

n!/(n-0)! = 1 for every n: there is exactly one arrangement of nothing. Microsoft does not print this row, but the value is forced by the equation the page does print, and the page's error list explicitly permits number_chosen = 0 (it errors only "If number <= 0 or if number_chosen < 0"). The empty permutation is the classic off-by-one in a hand-rolled implementation.

Microsoft documents: "Both arguments are truncated to integers." Truncation (not rounding) of 3.9 and 2.9 gives 3 and 2, so this must return PERMUT(3,2) = 6. An engine that ROUNDED instead would compute PERMUT(4,3) = 24 -- a four-fold difference, which makes this a decisive one-value test of the documented rule.

Microsoft documents: "If number < number_chosen, PERMUT returns the #NUM! error value."

Microsoft documents: "If number <= 0 or if number_chosen < 0, PERMUT returns the #NUM! error value." Zero is the boundary the <= includes, so PERMUT(0,0) is an error even though the empty arrangement of an empty set is mathematically 1 -- and note that PERMUTATIONA, the sibling function, documents the opposite treatment of the same input. That disagreement between two Microsoft pages is itself worth pinning. EXECUTED RESULT -- A SILENT WRONG ANSWER: LibreOffice returns 1 on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3), which agree case for case, where Excel documents #NUM!. Mathematically 1 is defensible -- there is exactly one arrangement of nothing -- but it is not what Microsoft's page specifies, and the disagreement is invisible: no error, no warning, just a number where Excel gives an error. Note that LibreOffice is here MORE permissive than Excel, the opposite direction from most of this batch's divergences.

Microsoft documents: "If number or number_chosen is nonnumeric, PERMUT returns the #VALUE! error value."

Microsoft publishes =PERMUT(A2,A3) = 970200 for 100 objects taken 3 at a time. DERIVATION in exact integer arithmetic from the documented equation n!/(n-k)!: 100 x 99 x 98 = 970200 exactly. No factorial of 100 is ever formed -- the ratio telescopes to the product of the top k terms, which is both the honest reading of the formula and the only way to evaluate it without overflow. Microsoft's page was re-read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/permut-function (the /en-us/office/<name>-function-<guid> path was serving Microsoft's 'Sorry, the page you're looking for can't be found' body throughout this batch, and the working path returns an 87 KB stub about half the time, so pages were fetched with retries until the payload exceeded 150 KB).

Microsoft publishes =PERMUT(3,2) = 6, "Permutations possible for a group of 3 objects where 2 are chosen". 3!/1! = 3 x 2 = 6, and the six orderings can be written out by hand, so this case is verifiable without arithmetic at all.

n!/(n-0)! = 1 for every n: there is exactly one arrangement of nothing. Microsoft does not print this row, but the value is forced by the equation the page does print, and the page's error list explicitly permits number_chosen = 0 (it errors only "If number <= 0 or if number_chosen < 0"). The empty permutation is the classic off-by-one in a hand-rolled implementation.

Microsoft documents: "Both arguments are truncated to integers." Truncation (not rounding) of 3.9 and 2.9 gives 3 and 2, so this must return PERMUT(3,2) = 6. An engine that ROUNDED instead would compute PERMUT(4,3) = 24 -- a four-fold difference, which makes this a decisive one-value test of the documented rule.

Microsoft documents: "If number < number_chosen, PERMUT returns the #NUM! error value."

Microsoft documents: "If number <= 0 or if number_chosen < 0, PERMUT returns the #NUM! error value." Zero is the boundary the <= includes, so PERMUT(0,0) is an error even though the empty arrangement of an empty set is mathematically 1 -- and note that PERMUTATIONA, the sibling function, documents the opposite treatment of the same input. That disagreement between two Microsoft pages is itself worth pinning. EXECUTED RESULT -- A SILENT WRONG ANSWER: LibreOffice returns 1 on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3), which agree case for case, where Excel documents #NUM!. Mathematically 1 is defensible -- there is exactly one arrangement of nothing -- but it is not what Microsoft's page specifies, and the disagreement is invisible: no error, no warning, just a number where Excel gives an error. Note that LibreOffice is here MORE permissive than Excel, the opposite direction from most of this batch's divergences.

Microsoft documents: "If number or number_chosen is nonnumeric, PERMUT returns the #VALUE! error value."

Microsoft publishes =PERMUT(A2,A3) = 970200 for 100 objects taken 3 at a time. DERIVATION in exact integer arithmetic from the documented equation n!/(n-k)!: 100 x 99 x 98 = 970200 exactly. No factorial of 100 is ever formed -- the ratio telescopes to the product of the top k terms, which is both the honest reading of the formula and the only way to evaluate it without overflow. Microsoft's page was re-read live on 2026-08-31 at https://support.microsoft.com/en-us/excel/functions/permut-function (the /en-us/office/<name>-function-<guid> path was serving Microsoft's 'Sorry, the page you're looking for can't be found' body throughout this batch, and the working path returns an 87 KB stub about half the time, so pages were fetched with retries until the payload exceeded 150 KB).

Microsoft publishes =PERMUT(3,2) = 6, "Permutations possible for a group of 3 objects where 2 are chosen". 3!/1! = 3 x 2 = 6, and the six orderings can be written out by hand, so this case is verifiable without arithmetic at all.

n!/(n-0)! = 1 for every n: there is exactly one arrangement of nothing. Microsoft does not print this row, but the value is forced by the equation the page does print, and the page's error list explicitly permits number_chosen = 0 (it errors only "If number <= 0 or if number_chosen < 0"). The empty permutation is the classic off-by-one in a hand-rolled implementation.

Microsoft documents: "Both arguments are truncated to integers." Truncation (not rounding) of 3.9 and 2.9 gives 3 and 2, so this must return PERMUT(3,2) = 6. An engine that ROUNDED instead would compute PERMUT(4,3) = 24 -- a four-fold difference, which makes this a decisive one-value test of the documented rule.

Microsoft documents: "If number < number_chosen, PERMUT returns the #NUM! error value."

Microsoft documents: "If number <= 0 or if number_chosen < 0, PERMUT returns the #NUM! error value." Zero is the boundary the <= includes, so PERMUT(0,0) is an error even though the empty arrangement of an empty set is mathematically 1 -- and note that PERMUTATIONA, the sibling function, documents the opposite treatment of the same input. That disagreement between two Microsoft pages is itself worth pinning. EXECUTED RESULT -- A SILENT WRONG ANSWER: LibreOffice returns 1 on all four pinned builds (24.2.0.3, 24.8.7.2, 25.2.0.3, 25.8.7.3), which agree case for case, where Excel documents #NUM!. Mathematically 1 is defensible -- there is exactly one arrangement of nothing -- but it is not what Microsoft's page specifies, and the disagreement is invisible: no error, no warning, just a number where Excel gives an error. Note that LibreOffice is here MORE permissive than Excel, the opposite direction from most of this batch's divergences.

Microsoft documents: "If number or number_chosen is nonnumeric, PERMUT returns the #VALUE! error value."