[iaspltxo] Some small binomial and permutation coefficients, sorted

We consider the numbers of permutations and combinations of a small number of objects.

permutation(n,k) = binomial(n,k)*k!

We omit from consideration permutation(n,n) because it is the same as permutation(n,n-1).  (Both are equal to n!.)  We omit from consideration binomial(n,n-k) because of symmetry.  We omit from consideration permutation(n,1), binomial(n,0), and binomial(n,1), because they are boring.

After those are omitted, there still remain multiple ways to achieve the values 6 20 56 120 210 462 720 990....

Binomial coefficients separated by small gaps (e.g., 495 and 496) might also be interesting to investigate.  Previously, twin smooth composites.

Inspired by the NBA draft lottery, which used (14 choose 4) = 1001 as a d1000 per mille randomizer, redrawing if the one unassigned combination was drawn.  Generally useful for this kind of task would be values equal to or just larger than a perfect power.

6 = binomial 4 2
6 = permutation 3 2
10 = binomial 5 2
12 = permutation 4 2
15 = binomial 6 2
20 = binomial 6 3
20 = permutation 5 2
21 = binomial 7 2
24 = permutation 4 3
28 = binomial 8 2
30 = permutation 6 2
35 = binomial 7 3
36 = binomial 9 2
42 = permutation 7 2
45 = binomial 10 2
55 = binomial 11 2
56 = binomial 8 3
56 = permutation 8 2
60 = permutation 5 3
66 = binomial 12 2
70 = binomial 8 4
72 = permutation 9 2
78 = binomial 13 2
84 = binomial 9 3
90 = permutation 10 2
91 = binomial 14 2
105 = binomial 15 2
110 = permutation 11 2
120 = binomial 10 3
120 = binomial 16 2
120 = permutation 5 4
120 = permutation 6 3
126 = binomial 9 4
132 = permutation 12 2
136 = binomial 17 2
153 = binomial 18 2
156 = permutation 13 2
165 = binomial 11 3
171 = binomial 19 2
182 = permutation 14 2
190 = binomial 20 2
210 = binomial 10 4
210 = binomial 21 2
210 = permutation 15 2
210 = permutation 7 3
220 = binomial 12 3
231 = binomial 22 2
240 = permutation 16 2
252 = binomial 10 5
253 = binomial 23 2
272 = permutation 17 2
276 = binomial 24 2
286 = binomial 13 3
300 = binomial 25 2
306 = permutation 18 2
325 = binomial 26 2
330 = binomial 11 4
336 = permutation 8 3
342 = permutation 19 2
351 = binomial 27 2
360 = permutation 6 4
364 = binomial 14 3
378 = binomial 28 2
380 = permutation 20 2
406 = binomial 29 2
420 = permutation 21 2
435 = binomial 30 2
455 = binomial 15 3
462 = binomial 11 5
462 = permutation 22 2
465 = binomial 31 2
495 = binomial 12 4
496 = binomial 32 2
504 = permutation 9 3
506 = permutation 23 2
528 = binomial 33 2
552 = permutation 24 2
560 = binomial 16 3
561 = binomial 34 2
595 = binomial 35 2
600 = permutation 25 2
630 = binomial 36 2
650 = permutation 26 2
666 = binomial 37 2
680 = binomial 17 3
702 = permutation 27 2
703 = binomial 38 2
715 = binomial 13 4
720 = permutation 10 3
720 = permutation 6 5
741 = binomial 39 2
756 = permutation 28 2
780 = binomial 40 2
792 = binomial 12 5
812 = permutation 29 2
816 = binomial 18 3
820 = binomial 41 2
840 = permutation 7 4
861 = binomial 42 2
870 = permutation 30 2
903 = binomial 43 2
924 = binomial 12 6
930 = permutation 31 2
946 = binomial 44 2
969 = binomial 19 3
990 = binomial 45 2
990 = permutation 11 3
992 = permutation 32 2
1001 = binomial 14 4
1035 = binomial 46 2
1056 = permutation 33 2
1081 = binomial 47 2

This list complete up to 1081.