# 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?

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

1. Anagrams

Binomial Anagrams
Triangular
 269C2 = 36046 296C2 = 43660 (EF) 496C2 = 122760 649C2 = 210276 (EF) 1973C2 = 1945378 3791C2 = 7183945 (JD) 2021C2 = 2041210 2201C2 = 2421100 (EF) 2022C2 = 2043231 2202C2 = 2423301 (EF) 2023C2 = 2045253 2203C2 = 2425503 (EF) 2024C2 = 2047276 2204C2 = 2427706 (EF) 2031C2 = 2061465 2301C2 = 2646150 (EF) * 2041C2 = 2081820 2401C2 = 2881200 (EF) * 2042C2 = 2083861 2402C2 = 2883601 (EF) 2043C2 = 2085903 4023C2 = 8090253 (JD) 2044C2 = 2087946 4024C2 = 8094276 (JD) 2157C2 = 2325246 2175C2 = 2364225 (EF) 2346C2 = 2750685 3426C2 = 5867025 (EF) 2365C2 = 2795430 2635C2 = 3470295 (EF) 2429C2 = 2948806 4292C2 = 9208486 (JD) 2458C2 = 3019653 4258C2 = 9063153 (JD) 2580C2 = 3326910 2805C2 = 3932610 (EF) 2847C2 = 4051281 2874C2 = 4128501 (EF) 3060C2 = 4680270 3600C2 = 6478200 (EF) 3068C2 = 4704778 3860C2 = 7447870 (JP) 3231C2 = 5218065 3321C2 = 5512860 (EF) 3403C2 = 5788503 3430C2 = 5880735 (EF) 3410C2 = 5812345 4103C2 = 8415253 (JP) 3435C2 = 5897895 3453C2 = 5959878 (EF) 3496C2 = 6109260 4396C2 = 9660210 (JP) 3572C2 = 6377806 3752C2 = 7036876 (JP) 3672C2 = 6739956 3726C2 = 6939675 (JP) 3679C2 = 6765681 3967C2 = 7866561 (JP) 4020C2 = 8078190 4200C2 = 8817900 (JP) * 4021C2 = 8082210 4201C2 = 8822100 (JP) * 4022C2 = 8086231 4202C2 = 8826301 (JD) 4074C2 = 8296701 4407C2 = 9708621 (JP) 4348C2 = 9450378 4438C2 = 9845703 (JP) 4415C2 = 9743905 4451C2 = 9903475 (JP) 4427C2 = 9796951 4472C2 = 9997156 (JP) 4506C2 = 10149765 5406C2 = 14609715 (JP) 4569C2 = 10435596 9654C2 = 46595031 (JP) 4679C2 = 10944181 4796C2 = 11498410 (JP) 4688C2 = 10986328 8864C2 = 39280816 (JP) 4752C2 = 11288376 5247C2 = 13762881 (JP) 4972C2 = 12357906 7429C2 = 27591306 (JP) 5005C2 = 12522510 5500C2 = 15122250 (JP) 5027C2 = 12632851 7502C2 = 28136251 (JP)

 5041C2 = 12703320 5140C2 = 13207230 (JP) 5042C2 = 12708361 5240C2 = 13726180 (JP) 5280C2 = 13936560 8205C2 = 33656910 (JP) 5336C2 = 14233780 5363C2 = 14378203 (JP) 5372C2 = 14426506 5732C2 = 16425046 (JP) 5375C2 = 14442625 5735C2 = 16442245 (JP) 5378C2 = 14458753 8735C2 = 38145745 (JP) 5635C2 = 15873795 5653C2 = 15975378 (JP) 5642C2 = 15913261 6542C2 = 21395611 (JP) 5698C2 = 16230753 6598C2 = 21763503 (JP) 5738C2 = 16459453 8357C2 = 34915546 (JP) 5970C2 = 17817465 9507C2 = 45186771 (JP) 5989C2 = 17931066 8599C2 = 36967101 (JP) 6179C2 = 19086931 6197C2 = 19198306 (JP) 6266C2 = 19628245 6626C2 = 21948625 (JP) 6298C2 = 19829253 6928C2 = 23995128 (JP) 6340C2 = 20094630 6403C2 = 20496003 (JP) 6358C2 = 20208903 6385C2 = 20380920 (JP) 6765C2 = 22879230 7665C2 = 29372280 (JP) 6853C2 = 23478378 8653C2 = 37432878 (JP) 7031C2 = 24713965 7310C2 = 26714395 (JP) 7128C2 = 25400628 7281C2 = 26502840 (JP) 7138C2 = 25471953 7183C2 = 25794153 (JP) 7191C2 = 25851645 9117C2 = 41555286 (JP) 7384C2 = 27258036 7438C2 = 27658203 (JP) 7817C2 = 30548836 8771C2 = 38460835 (JP) 7879C2 = 31035381 8779C2 = 38531031 (JP) 7901C2 = 31208950 7910C2 = 31280095 (JP) 7980C2 = 31836210 8079C2 = 32631081 (JP) 8018C2 = 32140153 8180C2 = 33452110 (JP) 8153C2 = 33231628 8513C2 = 36231328 (JP) 8195C2 = 33574915 8915C2 = 39734155 (JP) 8379C2 = 35099631 8937C2 = 39930516 (JP) 8385C2 = 35149920 8835C2 = 39024195 (JP) 8539C2 = 36452991 8935C2 = 39912645 (JP) 8749C2 = 38268126 8794C2 = 38662821 (JP) 9036C2 = 40820130 9360C2 = 43800120 (JP) 9112C2 = 41509716 9121C2 = 41591760 (JP) 9230C2 = 42591835 9302C2 = 43258951 (JP) 9547C2 = 45567831 9754C2 = 47565381 (JP)

Tetrahedral
 406C3 = 11071620 460C3 = 16117020 (EF) 567C3 = 30220155 675C3 = 51030225 (EF) 728C3 = 64039976 782C3 = 79396460 (EF) 10181C3 = 175829636970 18011C3 = 973620897165 (JP) 10306C3 = 182386518720 10360C3 = 185268781320 (JP) 11225C3 = 235663172700 12251C3 = 306377602125 (JP) 11423C3 = 248356318671 13124C3 = 376658813124 (JP) 11575C3 = 258403969275 17515C3 = 895374629205 (JP) 11722C3 = 268375424440 17122C3 = 836442574240 (JP) 15061C3 = 569277033490 15601C3 = 632735997400 (JP) 17436C3 = 883312961740 17463C3 = 887423039611 (JP) 22365C3 = 1864220157030 32652C3 = 5801472126300 (JP) 22600C3 = 1923607294200 26002C3 = 2929671342000 (JP) 24318C3 = 2396507173116 28341C3 = 3793571612610 (JP) 25285C3 = 2693928680170 25582C3 = 2789981362060 (JP) 25863C3 = 2882936463011 28563C3 = 3883422061961 (JP) 26934C3 = 3256139046884 29634C3 = 4336862091584 (JP) 29465C3 = 4263083769980 29546C3 = 4298339670680 (JP) 31198C3 = 5060427975196 31981C3 = 5451099726670 (JP) 31619C3 = 5268074821569 31916C3 = 5417928825660 (JP) 35024C3 = 7159930087024 35402C3 = 7394270591800 (JP) 41243C3 = 11691437250541 42431C3 = 12731156144095 (JP) 42816C3 = 13080868941760 48216C3 = 18680791048360 (JP)

 43054C3 = 13300225577804 44350C3 = 14537852032700 (JP) 43520C3 = 13736796720640 50432C3 = 21376740693760 (JP) 43582C3 = 13795591408060 43852C3 = 14053589196700 (JP) 45248C3 = 15438962706496 52484C3 = 24093766951684 (JP) 46075C3 = 16301084674525 64705C3 = 45148376520160 (JP) 46579C3 = 16841906774329 75469C3 = 71636980471294 (JP) 47952C3 = 18375609596400 74925C3 = 70098966514350 (JP) 54269C3 = 26636686447094 54296C3 = 26676463944680 (JP) 54678C3 = 27243492588676 57864C3 = 32288776529464 (JP) 54969C3 = 27680794812044 54996C3 = 27721604844980 (JP) 56636C3 = 30276346298540 66356C3 = 48693356270420 (JP) 57563C3 = 31787500128461 75356C3 = 71315670881420 (JP) 64965C3 = 45694825434730 65694C3 = 47250459383644 (JP) 71483C3 = 60874980486781 78314C3 = 80047971888664 (JP) 72625C3 = 63839466019000 72652C3 = 63910694806300 (JP) 74581C3 = 69137852639570 81547C3 = 90376753812965 (JP) 74605C3 = 69204620237710 75604C3 = 72022109367604 (JP) 89077C3 = 117796088423650 98770C3 = 160587123798640 (JP) 90748C3 = 124550529861996 98470C3 = 159128266599540 (JP) 92759C3 = 133015691848459 97295C3 = 153499486031815 (JP) 93237C3 = 135082674039170 93372C3 = 135670298301740 (JP) 94987C3 = 142832667627045 99874C3 = 166032472758624 (JP) 97690C3 = 155376645632680 97960C3 = 156668533674520 (JP)

Higher-Dimensional
 3459C4 = 5954392001376 3549C4 = 6599002433751 (EF) 5604C4 = 41050285745001 6405C4 = 70058105214405 (JP) 14370C4 = 1775965325761320 17340C4 = 3765605727219315 (JP) 15081C4 = 2154450498189990 18501C4 = 4880099954491125 (JP) 23362C4 = 12408443067522320 26332C4 = 20027428106423435 (JP) 23718C4 = 13182298672431735 31278C4 = 39871387263114225 (JP) 25953C4 = 18898990965961800 35952C4 = 69599881018896900 (JP) 32438C4 = 46123759884753915 34382C4 = 58215349746193785 (JP) 41874C4 = 128086784325921876 48417C4 = 228942511637878680 (JP) 43895C4 = 154664131262038095 49835C4 = 256965204381631410 (JP) 45178C4 = 173555780642193950 47581C4 = 213534598707590615 (JP) 55971C4 = 408878680724913960 57915C4 = 468714087968802390 (JP) 71970C4 = 1117785733041442920 77091C4 = 1471531874027134920 (JP) 80619C4 = 1759973234802562626 80916C4 = 1786052322269937645 (JP) 84708C4 = 2145140329578536745 84780C4 = 2152443057837419655 (JP) 86277C4 = 2308546876744361925 87726C4 = 2467586379634801425 (JP) 90717C4 = 2821725433467881655 91077C4 = 2866784283357141525 (JP) 95385C4 = 3448909318502571210 95538C4 = 3471091525280318940 (JP) 17182C5 = 12471955254541380906 17812C5 = 14932700146958145552 (JP) 44472C5 = 1449287011564979898024 44724C5 = 1490818725997604128944 (JP) 54829C5 = 4128489346207799017215 58942C5 = 5927470631709142814298 (JP) 74878C5 = 19612453714517490929850 88477C5 = 45177439212918901406595 (JP) 20932C6 = 116740248815973722928352 23209C6 = 216932872817759232815044 (JP) 33864C6 = 2093652769554893663938108 36483C6 = 3273649609135269305569888 (JP) 50956C6 = 24305895626756194080607516 59650C6 = 62549087386952617461505600 (JP) 74139C6 = 230600357969611548547443178 79431C6 = 348759197003628665447504311 (JP) 65276C7 = 1001623436649759159123627825800 75626C7 = 2806381939152477521326614695000 (JP) 73682C7 = 2338705420011603297284842961936 87263C7 = 7643212308345679690028224803191 (JP) 74573C7 = 2544007949713621856106493377812 77534C7 = 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.