Problem of the Month (October 2011)

This month we investigate several questions about binomial coefficients nCr = n! / r! (n-r)! .

1. Anagrams

In some cases, permuting the digits of n just permutes the digits of nCr. For example, 269C2 = 36046 and 296C2 = 43660. How many more examples are there of this type?

2. Sums

There are lots of examples of triangular numbers that are the sum of two other triangular numbers, and tetrahedral numbers that are the sum of two tetrahedral numbers. What about other binomial coefficients? The only example known besides the trivial family 2nCn = 2n-1Cn + 2n-1Cn is 200C4 = 190C4 + 132C4. Are there any other examples?

3. Products

There are lots of examples of triangular numbers that are the product of two other triangular numbers. What about other binomial coefficients? For example, 16C3 = 8C3 × 5C3. How many more examples are there of this type?

4. Printer Errors

In some cases, if a printer forgets to subscript some digits, the result is unchanged. For example, 215C2 = 215C2. Are there any other examples?

5. Concatenation

There are lots of examples of triangular numbers that are the concatenation of other triangular numbers, and tetrahedral numbers that are the concatenation of tetrahedral numbers. What about other binomial coefficients? For example, 1748C4 = 387670507005 is the concatenation of 3876,70,5,0,70,0,5, all of which are nC4 for some n. How many more examples are there of this type?

6. Factorials

What is the fewest number of non-trivial binomial coefficients that we need to multiply in order to get factorials? For example, 5! = 10C3 and 6! = 10C3 × 4C2.

7. Mountains

An integer is called a mountain if the digits are non-decreasing followed by non-increasing. There are infinitely many triangular and tetrahedral numbers that are mountains. But there seem to be only finitely many mountains among nCk for any k≥4. What are the largest? Are there any bigger than 434C4=1457898876 and 38C16=22239974430?


ANSWERS

Solutions were received from Jon Palin, Johan de Ruiter, Mark Mammel, and Joe DeVincentis.

1. Anagrams

Binomial Anagrams
Triangular
269C2 = 36046296C2 = 43660(EF)
496C2 = 122760649C2 = 210276(EF)
1973C2 = 19453783791C2 = 7183945(JD)
2021C2 = 20412102201C2 = 2421100(EF)
2022C2 = 20432312202C2 = 2423301(EF)
2023C2 = 20452532203C2 = 2425503(EF)
2024C2 = 20472762204C2 = 2427706(EF)
2031C2 = 20614652301C2 = 2646150(EF)
* 2041C2 = 20818202401C2 = 2881200(EF)
* 2042C2 = 20838612402C2 = 2883601(EF)
2043C2 = 20859034023C2 = 8090253(JD)
2044C2 = 20879464024C2 = 8094276(JD)
2157C2 = 23252462175C2 = 2364225(EF)
2346C2 = 27506853426C2 = 5867025(EF)
2365C2 = 27954302635C2 = 3470295(EF)
2429C2 = 29488064292C2 = 9208486(JD)
2458C2 = 30196534258C2 = 9063153(JD)
2580C2 = 33269102805C2 = 3932610(EF)
2847C2 = 40512812874C2 = 4128501(EF)
3060C2 = 46802703600C2 = 6478200(EF)
3068C2 = 47047783860C2 = 7447870(JP)
3231C2 = 52180653321C2 = 5512860(EF)
3403C2 = 57885033430C2 = 5880735(EF)
3410C2 = 58123454103C2 = 8415253(JP)
3435C2 = 58978953453C2 = 5959878(EF)
3496C2 = 61092604396C2 = 9660210(JP)
3572C2 = 63778063752C2 = 7036876(JP)
3672C2 = 67399563726C2 = 6939675(JP)
3679C2 = 67656813967C2 = 7866561(JP)
4020C2 = 80781904200C2 = 8817900(JP)
* 4021C2 = 80822104201C2 = 8822100(JP)
* 4022C2 = 80862314202C2 = 8826301(JD)
4074C2 = 82967014407C2 = 9708621(JP)
4348C2 = 94503784438C2 = 9845703(JP)
4415C2 = 97439054451C2 = 9903475(JP)
4427C2 = 97969514472C2 = 9997156(JP)
4506C2 = 101497655406C2 = 14609715(JP)
4569C2 = 104355969654C2 = 46595031(JP)
4679C2 = 109441814796C2 = 11498410(JP)
4688C2 = 109863288864C2 = 39280816(JP)
4752C2 = 112883765247C2 = 13762881(JP)
4972C2 = 123579067429C2 = 27591306(JP)
5005C2 = 125225105500C2 = 15122250(JP)
5027C2 = 126328517502C2 = 28136251(JP)

5041C2 = 127033205140C2 = 13207230(JP)
5042C2 = 127083615240C2 = 13726180(JP)
5280C2 = 139365608205C2 = 33656910(JP)
5336C2 = 142337805363C2 = 14378203(JP)
5372C2 = 144265065732C2 = 16425046(JP)
5375C2 = 144426255735C2 = 16442245(JP)
5378C2 = 144587538735C2 = 38145745(JP)
5635C2 = 158737955653C2 = 15975378(JP)
5642C2 = 159132616542C2 = 21395611(JP)
5698C2 = 162307536598C2 = 21763503(JP)
5738C2 = 164594538357C2 = 34915546(JP)
5970C2 = 178174659507C2 = 45186771(JP)
5989C2 = 179310668599C2 = 36967101(JP)
6179C2 = 190869316197C2 = 19198306(JP)
6266C2 = 196282456626C2 = 21948625(JP)
6298C2 = 198292536928C2 = 23995128(JP)
6340C2 = 200946306403C2 = 20496003(JP)
6358C2 = 202089036385C2 = 20380920(JP)
6765C2 = 228792307665C2 = 29372280(JP)
6853C2 = 234783788653C2 = 37432878(JP)
7031C2 = 247139657310C2 = 26714395(JP)
7128C2 = 254006287281C2 = 26502840(JP)
7138C2 = 254719537183C2 = 25794153(JP)
7191C2 = 258516459117C2 = 41555286(JP)
7384C2 = 272580367438C2 = 27658203(JP)
7817C2 = 305488368771C2 = 38460835(JP)
7879C2 = 310353818779C2 = 38531031(JP)
7901C2 = 312089507910C2 = 31280095(JP)
7980C2 = 318362108079C2 = 32631081(JP)
8018C2 = 321401538180C2 = 33452110(JP)
8153C2 = 332316288513C2 = 36231328(JP)
8195C2 = 335749158915C2 = 39734155(JP)
8379C2 = 350996318937C2 = 39930516(JP)
8385C2 = 351499208835C2 = 39024195(JP)
8539C2 = 364529918935C2 = 39912645(JP)
8749C2 = 382681268794C2 = 38662821(JP)
9036C2 = 408201309360C2 = 43800120(JP)
9112C2 = 415097169121C2 = 41591760(JP)
9230C2 = 425918359302C2 = 43258951(JP)
9547C2 = 455678319754C2 = 47565381(JP)

Tetrahedral
406C3 = 11071620460C3 = 16117020(EF)
567C3 = 30220155675C3 = 51030225(EF)
728C3 = 64039976782C3 = 79396460(EF)
10181C3 = 17582963697018011C3 = 973620897165(JP)
10306C3 = 18238651872010360C3 = 185268781320(JP)
11225C3 = 23566317270012251C3 = 306377602125(JP)
11423C3 = 24835631867113124C3 = 376658813124(JP)
11575C3 = 25840396927517515C3 = 895374629205(JP)
11722C3 = 26837542444017122C3 = 836442574240(JP)
15061C3 = 56927703349015601C3 = 632735997400(JP)
17436C3 = 88331296174017463C3 = 887423039611(JP)
22365C3 = 186422015703032652C3 = 5801472126300(JP)
22600C3 = 192360729420026002C3 = 2929671342000(JP)
24318C3 = 239650717311628341C3 = 3793571612610(JP)
25285C3 = 269392868017025582C3 = 2789981362060(JP)
25863C3 = 288293646301128563C3 = 3883422061961(JP)
26934C3 = 325613904688429634C3 = 4336862091584(JP)
29465C3 = 426308376998029546C3 = 4298339670680(JP)
31198C3 = 506042797519631981C3 = 5451099726670(JP)
31619C3 = 526807482156931916C3 = 5417928825660(JP)
35024C3 = 715993008702435402C3 = 7394270591800(JP)
41243C3 = 1169143725054142431C3 = 12731156144095(JP)
42816C3 = 1308086894176048216C3 = 18680791048360(JP)

43054C3 = 1330022557780444350C3 = 14537852032700(JP)
43520C3 = 1373679672064050432C3 = 21376740693760(JP)
43582C3 = 1379559140806043852C3 = 14053589196700(JP)
45248C3 = 1543896270649652484C3 = 24093766951684(JP)
46075C3 = 1630108467452564705C3 = 45148376520160(JP)
46579C3 = 1684190677432975469C3 = 71636980471294(JP)
47952C3 = 1837560959640074925C3 = 70098966514350(JP)
54269C3 = 2663668644709454296C3 = 26676463944680(JP)
54678C3 = 2724349258867657864C3 = 32288776529464(JP)
54969C3 = 2768079481204454996C3 = 27721604844980(JP)
56636C3 = 3027634629854066356C3 = 48693356270420(JP)
57563C3 = 3178750012846175356C3 = 71315670881420(JP)
64965C3 = 4569482543473065694C3 = 47250459383644(JP)
71483C3 = 6087498048678178314C3 = 80047971888664(JP)
72625C3 = 6383946601900072652C3 = 63910694806300(JP)
74581C3 = 6913785263957081547C3 = 90376753812965(JP)
74605C3 = 6920462023771075604C3 = 72022109367604(JP)
89077C3 = 11779608842365098770C3 = 160587123798640(JP)
90748C3 = 12455052986199698470C3 = 159128266599540(JP)
92759C3 = 13301569184845997295C3 = 153499486031815(JP)
93237C3 = 13508267403917093372C3 = 135670298301740(JP)
94987C3 = 14283266762704599874C3 = 166032472758624(JP)
97690C3 = 15537664563268097960C3 = 156668533674520(JP)

Higher-Dimensional
3459C4 = 59543920013763549C4 = 6599002433751(EF)
5604C4 = 410502857450016405C4 = 70058105214405(JP)
14370C4 = 177596532576132017340C4 = 3765605727219315(JP)
15081C4 = 215445049818999018501C4 = 4880099954491125(JP)
23362C4 = 1240844306752232026332C4 = 20027428106423435(JP)
23718C4 = 1318229867243173531278C4 = 39871387263114225(JP)
25953C4 = 1889899096596180035952C4 = 69599881018896900(JP)
32438C4 = 4612375988475391534382C4 = 58215349746193785(JP)
41874C4 = 12808678432592187648417C4 = 228942511637878680(JP)
43895C4 = 15466413126203809549835C4 = 256965204381631410(JP)
45178C4 = 17355578064219395047581C4 = 213534598707590615(JP)
55971C4 = 40887868072491396057915C4 = 468714087968802390(JP)
71970C4 = 111778573304144292077091C4 = 1471531874027134920(JP)
80619C4 = 175997323480256262680916C4 = 1786052322269937645(JP)
84708C4 = 214514032957853674584780C4 = 2152443057837419655(JP)
86277C4 = 230854687674436192587726C4 = 2467586379634801425(JP)
90717C4 = 282172543346788165591077C4 = 2866784283357141525(JP)
95385C4 = 344890931850257121095538C4 = 3471091525280318940(JP)
17182C5 = 1247195525454138090617812C5 = 14932700146958145552(JP)
44472C5 = 144928701156497989802444724C5 = 1490818725997604128944(JP)
54829C5 = 412848934620779901721558942C5 = 5927470631709142814298(JP)
74878C5 = 1961245371451749092985088477C5 = 45177439212918901406595(JP)
20932C6 = 11674024881597372292835223209C6 = 216932872817759232815044(JP)
33864C6 = 209365276955489366393810836483C6 = 3273649609135269305569888(JP)
50956C6 = 2430589562675619408060751659650C6 = 62549087386952617461505600(JP)
74139C6 = 23060035796961154854744317879431C6 = 348759197003628665447504311(JP)
65276C7 = 100162343664975915912362782580075626C7 = 2806381939152477521326614695000(JP)
73682C7 = 233870542001160329728484296193687263C7 = 7643212308345679690028224803191(JP)
74573C7 = 254400794971362185610649337781277534C7 = 3341159138790247640578427902616(JP)

Johan de Riuter found some infinite families of these.

Joe DeVincentis notes that the triangular solutions marked with a star can be combined.

Joe DeVincentis found these triples of anagrammed triangular numbers:

[10193, 10391, 10913] [51943528, 53981245, 59541328]
[20011, 20101, 21001] [200210055, 202015050, 220510500]
[20041, 24001, 40021, 42001] [200810820, 288012000, 800820210, 882021000]
[20042, 24002, 40022, 42002] [200830861, 288036001, 800860231, 882063001]
[20434, 23404, 40234] [208763961, 273861906, 809367261]
[22199, 29192, 29219] [246386701, 426071836, 426860371]
[24503, 40325, 40523] [300186253, 813032650, 821036503]
[30238, 30823, 38302] [457153203, 475013253, 733502451]
[40011, 40101, 41001] [800420055, 804025050, 840520500]
[45026, 56042, 60524] [1013647825, 1570324861, 1831547026]
[47255, 52574, 52754] [1116493885, 1381986451, 1391465881]
[49829, 89924, 92849] [1241439706, 4043117926, 4310421976]
[50015, 50501, 55001] [1250725105, 1275150250, 1512527500]
[50695, 55069, 65095] [1284966165, 1516269846, 2118646965]
[51269, 65291, 69512] [1314229546, 2131424695, 2415924316]
[51948, 59148, 89154] [1349271378, 1749213378, 3974173281]
[52499, 92954, 95942] [1378046251, 4320176581, 4602385711]
[53077, 70357, 70735] [1408557426, 2475018546, 2501684745]
[53172, 57132, 57312] [1413604206, 1632004146, 1642304016]
[53395, 53953, 55393] [1425486315, 1455436128, 1534164528]
[53725, 57325, 57523] [1443160950, 1643049150, 1654419003]
[57906, 95670, 97506] [1676523465, 4576326615, 4753661265]
[57938, 59783, 79835] [1678376953, 1786973653, 3186773695]
[60011, 60101, 61001] [1800630055, 1806035050, 1860530500]
[63654, 64536, 64563] [2025884031, 2082415380, 2084158203]
[67072, 67720, 70276] [2249293056, 2292965340, 2469322950]
[75281, 78152, 87251] [2833576840, 3053828476, 3806324875]
[77806, 86077, 87670] [3026847915, 3704581926, 3842970615]
[80011, 80101, 81001] [3200840055, 3208045050, 3280540500]
[89165, 89615, 95186] [3975154030, 4015379305, 4530139705]
[96429, 96924, 99624] [4649227806, 4697082426, 4962420876]

He also found this triple for n=4:
[913596, 919356, 969153] [29027034145968417206835, 29766020942103458487315, 36758302495182914427600]


2. Sums

Binomial Sums
16C6 = 15C6 + 14C6 (MM)
21C6 = 19C6 + 19C6(MM)
200C4 = 190C4 + 132C4(EF)
120C35 = 118C35 + 118C35(JD)
105C40 = 104C40 + 103C40(JD)
697C204 = 695C204 + 695C204(JD)
715C273 = 714C273 + 713C273(JD)

Joe DeVincentis showed that there are 2 infinite families that arise from some Pell equations:
When 5r2-2r+1 is a square, n=(1+3r+√(5r2-2r+1))/2 makes nCr = n-1Cr + n-2Cr.
When 8r2+1 is a square, n=(1+4r+√(8r2+1))/2 makes nCr = n-2Cr + n-2Cr.


3. Products

Binomial Products
Tetrahedral
16C3 = 8C3 × 5C3(EF)
50C3 = 16C3 × 7C3(EF)
65C3 = 14C3 × 10C3(EF)
176C3 = 30C3 × 12C3(EF)
210C3 = 78C3 × 6C3(EF)
352C3 = 65C3 × 11C3(EF)
561C3 = 86C3 × 13C3(EF)
2576C3 = 105C3 × 46C3(JP)
5425C3 = 341C3 × 30C3(JP)
11440C3 = 496C3 × 43C3(JP)

Higher-Dimensional
28C4 = 15C4 × 6C4(EF)
9C5 = 7C5 × 6C5(JP)
16C11 = 14C11 × 12C11(JD)
25C19 = 23C19 × 20C19(JD)
36C29 = 34C219 × 30C29(JD)
50C34 = 47C34 × 35C34(JD)

Joe DeVincentis showed that there is an infinite family that arise from a Pell equation:
When 4r+5 is a square, n=(3+2r-√(4r+5))/2 makes nCr = r+1Cr × n-2Cr.


4. Printer Errors

Binomial Printer Errors
2 × 105 = 215C2 = 215C2 = 21 × 10

Joe DeVincentis did an extensive analysis to streamline the search, but did not discover any new solutions.


5. Concatenation

Binomial Concatenations
6C4 = 1,5(EF)
14C4 = 1,0,0,1(EF)
25C4 = 126,5,0(EF)
26C4 = 1,495,0(EF)
1748C4 = 3876,70,5,0,70,0,5(EF)
25C6 = 1,7,7,1,0,0(JP)
16C8 = 1287,0(EF)
10C9 = 1,0(EF)
29C9 = 10,0,1,5005(EF)
11C10 = 1,1(JP)
14C10 = 1,0,0,1(JP)
21C14 = 11628,0(JP)
20C18 = 19,0(JP)
25C22 = 23.0,0(JP)
30C27 = 406,0(JP)
40C27 = 1203322288,0(JP)
40C36 = 9139,0(JP)
50C45 = 211876,0(JP)
60C54 = 5006386,0(JP)
70C63 = 119877472,0(JP)
80C72 = 2898753715,0(JP)
78C75 = 76,0,76(JP)
90C81 = 70625252863,0(JP)
100C90 = 1731030945644,0(JP)

100C99 = 1,0,0(JP)
101C100 = 1,0,1(JP)
110C99 = 42634215112710,0(JD)
110C109 = 1,1,0(JP)
111C110 = 1,1,1(JP)
120C108 = 1054285955968882,0(JD)
130C117 = 26159486052576800,0(JD)
140C126 = 650961395024165664,0(JD)
142C140 = 1,0,0,1,1(JP)
200C198 = 199,0,0(JD)
300C297 = 44551,0,0(JD)
400C396 = 10507399,0,0(JD)
500C495 = 2552446876,0,0(JD)
600C594 = 631952326369,0,0(JD)
700C693 = 158557355818786,0,0(JD)
800C792 = 40174579682804359,0,0(JD)
1000C999 = 1,0,0,0(JD)
1001C1000 = 1,0,0,1(JD)
1010C1009 = 1,0,1,0(JD)
1011C1010 = 1,0,1,1(JD)
1100C1099 = 1,1,0,0(JD)
1101C1100 = 1,1,0,1(JD)
1110C1109 = 1,1,1,0(JD)
1111C1110 = 1,1,1,1(JD)


6. Factorials

Shortest-Known Binomial Factorials
3! = 4C2(EF)
5! = 10C3(EF)
6! = 10C3 × 4C2(EF)
7! = 10C5 × 6C3(EF)
8! = 64C2 × 6C3(EF)
9! = 10C3 × 9C3 × 9C2(EF)
10! = 10C5 × 10C3 × 10C3(EF)
11! = 33C2 × 25C2 × 10C5(EF)
12! = 64C2 × 25C2 × 12C5(EF)
13! = 33C2 × 28C5 × 10C3(EF)
14! = 65C3 × 64C2 × 45C2(EF)
15! = 81C2 × 66C4 × 16C3(EF)
16! = 8C4 × 64C2 × 81C3 × 66C3(JP)
17! = 4096C2 × 100C2 × 18C5(JR)
18! = 4096C2 × 1701C2 × 33C2(JR)
19! = 2432C3 × 225C2 × 64C2(JR)
20! = 36C5 × 513C2 × 10C5 × 625C2(JP)
21! = 4096C2 × 225C2 × 81C2 × 22C6(JR)
22! = 4096C2 × 1216C2 × 56C3 × 35C3(JR)
23! = 4096C2 × 385C2 × 460C2 × 57C4(JR)

24! = 4096C2 × 1540C2 × 576C2 × 36C5(JR)
25! = 165376C2 × 4096C2 × 2025C2 × 12C2(JR)
26! = 21505C2 × 4375C2 × 4096C2 × 209C2(JR)
27! = 76545C2 × 4096C2 × 2376C2 × 561C2(JR)
28! = 4375C2 × 4096C2 × 4096C2 × 24C10 × 22C2(JR)
29! = 165376C2 × 76545C2 × 7425C2 × 16C6(JR)
30! = 4096C2 × 4096C2 × 2025C2 × 1701C6 × 1596C2(JR)
31! = 3301376C2 × 20736C2 × 1216C2 × 136C2 × 46C2(JR)
32! = 4375C2 × 4096C2 × 4096C2 × 36C5 × 33C15(JR)
33! = 42688C2 × 2376C2 × 4096C2 × 4096C2 × 9801C2(JR)
34! = 165376C2 × 13312C2 × 4096C2 × 3565C2 × 3025C2(JR)
35! = 165376C2 × 76545C2 × 4096C2 × 3025C2 × 35C7(JR)
36! = 176001C2 × 76545C2 × 6480C2 × 4096C2 × 385C2 × 36C2(JR)
37! = 33264C2 × 21505C2 × 4375C2 × 4096C2 × 2432C3 × 16C3(JR)
38! = 78337C2 × 20736C2 × 4096C2 × 3025C2 × 1702C3 × 225C2(JR)
39! = 2598400C2 × 303601C2 × 123201C2 × 4096C2 × 64C2 × 56C3(JR)
40! = 165376C2 × 76545C2 × 4096C2 × 3025C2 × 40C17 × 34C3(JR)
41! = 10491040C2 × 282625C2 × 15873C2 × 9801C2 × 4096C2 × 25C2(JR)
42! = 1048576C2 × 2893401C2 × 244036C2 × 4096C2 × 441C2 × 225C2(JR)
43! = 36315136C2 × 18113536C2 × 176001C2 × 4096C2 × 351C2 × 8C4(JR)


7. Mountains

Infinite Families of Mountains
Triangular
10(0)C2=4(9)5(0)(EF)
10(6)7C2=56(8)71(1)(EF)
1(3)4C2=(8)91(1)(EF)
160(0)C2=127(9)20(0)(EF)
1(6)7C2=13(8)6(1)(EF)
2(0)C2=1(9)(0)(EF)
22(6)7C2=256(8)51(1)(EF)
2600(0)C2=337(9)8700(0)(EF)
2(6)7C2=35(5)1(1)(EF)
2(6)8C2=3(5)7(7)8(EF)
300(0)C2=44(9)850(0)(EF)
(3)4C2=(5)6(1)(EF)
3400(0)C2=577(9)8300(0)(EF)
40(0)C2=7(9)80(0)(EF)
500(0)C2=124(9)750(0)(EF)
5400(0)C2=1457(9)7300(0)(EF)
5600(0)C2=1567(9)7200(0)(EF)
60(0)C2=17(9)0(0)(EF)
(6)7C2=2(2)1(1)(EF)
(6)71C2=2(2)4(7)85(JP)
(6)8C2=2(2)(7)8(EF)
(6)684C2=2(2)33(4)586(JP)
(6)686C2=2(2)34(7)955(JP)
(6)9C2=(2)3(4)6(EF)
68(6)67C2=2357(5)4411(1)(JP)
700(0)C2=244(9)650(0)(EF)
74(6)67C2=2787(5)4311(1)(JP)
8(6)7C2=37(5)41(1)(EF)

Tetrahedral
1(0)1C3=16(66)5(0)
14(0)1C3=4573(33)10(0)
2(0)1C3=133(33)0(0)
3(0)1C3=449(99)5(0)
42(0)1C3=123479(99)0(0)
6(0)1C3=3599(99)0(0)
8(0)1C3=853(33)20(0)
88(0)1C3=113578(66)520(0)

Largest-Known Non-Trivial Binomial Mountains
130C4=11358880
145C4=17666220
50C7=99884400
51C7=115775100
434C4=1457898876
38C16=22239974430

Jon Palin suggested we also look for valleys.


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 10/1/11.