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Binomial Pandigital(s) (Posted on 2009-04-28) Difficulty: 2 of 5
Determine all possible pair(s) (M, X) of positive integers, with M - 1 > X > 1, such that the decimal representation of MCX consists of non leading zeroes and contains each of the digits 0 to 9 exactly once.

Note: MCX is the number of X-element subsets (the X-combinations) of an M-element set, that is the number of ways that X things can be 'chosen' from a set of M things.

See The Solution Submitted by K Sengupta    
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Solution computer solution in basic | Comment 2 of 3 |

The program will capture only the lower value of X for C(M,X) and C(M,M-X), as it will abandon a given M when the X's cause the combinations to go above a 10-digit number.

UBASIC will not calculate by itself C(M,X) when M > 65536, so this is done via the formula for X = 2. Further values of X are not required as C(65535,3) is already 46,908,201,271,295, a 14-digit number.

As C(141422,2) already exceeds 10 digits, we need go no farther with M.

Thus the results will be complete.

   10   Lm=141422
   20   for M=4 to Lm
   30     for X=2 to M-2
   35        if M>=65536 and X>2 then cancel for:goto *NotThis
   40        if M<65536 then C=combi(M,X):else C=int(M*(M-1)/2)
   50        if C>9999999999 then cancel for:goto *NotThis
   60        Cs=cutspc(str(C))
   65        Good=0
   70        if len(Cs)=10 then
   75        :Good=1
   80        :for I=1 to 9
   90         :if instr(I+1,Cs,mid(Cs,I,1))>0 then Good=0:endif
  100        :next
  110        if Good then print M;tab(12);X;tab(18);C:inc SCt
  120     next X
  125   *NotThis
  130   next M
  140   print SCt
 


There are 84 results; you could double this by setting X = M - X in each instance: 

  
 M          X       C(M,X)
253         5     8301429675
595         4     5169738420
46098       2     1062489753
49797       2     1239845706
50140       2     1256984730
55152       2     1520843976
55485       2     1539264870
56521       2     1597283460
58051       2     1684930275
62496       2     1952843760
62568       2     1957346028
62901       2     1978236450
66295       2     2197480365
68806       2     2367098415
69543       2     2418079653
70767       2     2503948761
72595       2     2634980715
73738       2     2718609453
73972       2     2735891406
74169       2     2750483196
74358       2     2764518903
75556       2     2854316790
76365       2     2915768430
77806       2     3026847915
78687       2     3095782641
78849       2     3108542976
84556       2     3574816290
85960       2     3694517820
86077       2     3704581926
87264       2     3807459216
87670       2     3842970615
88407       2     3907854621
89884       2     4039521786
90288       2     4075916328
90298       2     4076819253
90981       2     4138725690
91477       2     4183975026
91836       2     4216879530
93393       2     4361079528
93627       2     4382960751
95112       2     4523098716
96994       2     4703869521
97488       2     4751906328
97965       2     4798521630
98685       2     4869315270
98758       2     4876521903
99271       2     4927316085
99325       2     4932678150
99613       2     4961325078
100387      2     5038724691
100747      2     5074928631
101224      2     5123098476
101709      2     5172309486
104113      2     5419706328
104202      2     5428976301
104779      2     5489267031
107154      2     5740936281
107605      2     5789364210
107829      2     5813492706
109405      2     5984672310
110395      2     6093472815
112708      2     6351490278
114039      2     6502389741
117081      2     6853921740
117423      2     6894021753
118071      2     6970321485
120699      2     7284063951
121815      2     7419386205
122221      2     7468925310
122329      2     7482130956
125064      2     7820439516
125236      2     7841965230
126162      2     7958362041
128341      2     8235641970
130689      2     8539742016
131382      2     8630549271
133615      2     8926417305
133839      2     8956372041
133876      2     8961324750
134442      2     9037258461
135658      2     9201478653
138394      2     9576380421
138816      2     9634871520
138960      2     9654871320

  Posted by Charlie on 2009-04-28 12:28:47
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