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Last Updated: May 20, 2019

Probabilities for the Sum of 1 to 25 Dice

Introduction

This section endeavors to answer the frequently asked question on the probability for any given total over the throw of multiple dice. I shall show the number of combinations (up to 15 significant digits) and probability (to 15 decimal places) of totals for 1 to 10, 15, 20, and 25 dice. I also keep just the number of combinations for all numbers of dice from 1 to 26, without the 15 significant digit constraint, in a section for the unedited data.

Combinations for One to Six Dice.

The first table shows just the number of combinations for one to six dice.

Notice how for two or more dice the number of combinations equals the sum of combinations one column to the left, starting from one row higher to seven rows higher. For example, comb(12 with 3 dice)=comb(11 with 2 dice)+comb(10 with 2 dice)+comb(9 with 2 dice)+comb(8 with 2 dice)+comb(7 with 2 dice)+comb(6 with 2 dice), or 25=2+3+4+5+6+5. Assume the number of combinations for any total less than 1 is 0. This method will work for any number of dice.

Combinations for One to Six Dice

Total 1 die 2 dice 3 dice 4 dice 5 dice 6 dice
1 1
2 1 1
3 1 2 1
4 1 3 3 1
5 1 4 6 4 1
6 1 5 10 10 5 1
7 6 15 20 15 6
8 5 21 35 35 21
9 4 25 56 70 56
10 3 27 80 126 126
11 2 27 104 205 252
12 1 25 125 305 456
13 21 140 420 756
14 15 146 540 1161
15 10 140 651 1666
16 6 125 735 2247
17 3 104 780 2856
18 1 80 780 3431
19 56 735 3906
20 35 651 4221
21 20 540 4332
22 10 420 4221
23 4 305 3906
24 1 205 3431
25 126 2856
26 70 2247
27 35 1666
28 15 1161
29 5 756
30 1 456
31 252
32 126
33 56
34 21
35 6
36 1

1 Die

1 Die

Total Combinations Probability
1 1 0.166666666666667
2 1 0.166666666666667
3 1 0.166666666666667
4 1 0.166666666666667
5 1 0.166666666666667
6 1 0.166666666666667
Total 6 1.000000000000000

2 Dice

2 Dice

Total Combinations Probability
2 1 0.027777777777778
3 2 0.055555555555556
4 3 0.083333333333333
5 4 0.111111111111111
6 5 0.138888888888889
7 6 0.166666666666667
8 5 0.138888888888889
9 4 0.111111111111111
10 3 0.083333333333333
11 2 0.055555555555556
12 1 0.027777777777778
Total 36 1.000000000000000

3 Dice

3 Dice

Total Combinations Probability
3 1 0.004629629629630
4 3 0.013888888888889
5 6 0.027777777777778
6 10 0.046296296296296
7 15 0.069444444444444
8 21 0.097222222222222
9 25 0.115740740740741
10 27 0.125000000000000
11 27 0.125000000000000
12 25 0.115740740740741
13 21 0.097222222222222
14 15 0.069444444444444
15 10 0.046296296296296
16 6 0.027777777777778
17 3 0.013888888888889
18 1 0.004629629629630
Total 216 1.000000000000000

4 Dice

4 Dice

Total Combinations Probability
4 1 0.000771604938272
5 4 0.003086419753086
6 10 0.007716049382716
7 20 0.015432098765432
8 35 0.027006172839506
9 56 0.043209876543210
10 80 0.061728395061728
11 104 0.080246913580247
12 125 0.096450617283951
13 140 0.108024691358025
14 146 0.112654320987654
15 140 0.108024691358025
16 125 0.096450617283951
17 104 0.080246913580247
18 80 0.061728395061728
19 56 0.043209876543210
20 35 0.027006172839506
21 20 0.015432098765432
22 10 0.007716049382716
23 4 0.003086419753086
24 1 0.000771604938272
Total 1296 1.000000000000000

5 Dice

5 Dice

Total Combinations Probability
5 1 0.000128600823045
6 5 0.000643004115226
7 15 0.001929012345679
8 35 0.004501028806584
9 70 0.009002057613169
10 126 0.016203703703704
11 205 0.026363168724280
12 305 0.039223251028807
13 420 0.054012345679012
14 540 0.069444444444444
15 651 0.083719135802469
16 735 0.094521604938272
17 780 0.100308641975309
18 780 0.100308641975309
19 735 0.094521604938272
20 651 0.083719135802469
21 540 0.069444444444444
22 420 0.054012345679012
23 305 0.039223251028807
24 205 0.026363168724280
25 126 0.016203703703704
26 70 0.009002057613169
27 35 0.004501028806584
28 15 0.001929012345679
29 5 0.000643004115226
30 1 0.000128600823045
Total 7776 1.000000000000000

6 Dice

6 Dice

Total Combinations Probability
6 1 0.000021433470508
7 6 0.000128600823045
8 21 0.000450102880658
9 56 0.001200274348423
10 126 0.002700617283951
11 252 0.005401234567901
12 456 0.009773662551440
13 756 0.016203703703704
14 1161 0.024884259259259
15 1666 0.035708161865569
16 2247 0.048161008230453
17 2856 0.061213991769547
18 3431 0.073538237311386
19 3906 0.083719135802469
20 4221 0.090470679012346
21 4332 0.092849794238683
22 4221 0.090470679012346
23 3906 0.083719135802469
24 3431 0.073538237311386
25 2856 0.061213991769547
26 2247 0.048161008230453
27 1666 0.035708161865569
28 1161 0.024884259259259
29 756 0.016203703703704
30 456 0.009773662551440
31 252 0.005401234567901
32 126 0.002700617283951
33 56 0.001200274348423
34 21 0.000450102880658
35 6 0.000128600823045
36 1 0.000021433470508
Total 46656 1.000000000000000

7 Dice

7 Dice

Total Combinations Probability
7 1 0.000003572245085
8 7 0.000025005715592
9 28 0.000100022862369
10 84 0.000300068587106
11 210 0.000750171467764
12 462 0.001650377229081
13 917 0.003275748742570
14 1667 0.005954932556013
15 2807 0.010027291952446
16 4417 0.015778606538637
17 6538 0.023355338363054
18 9142 0.032657464563329
19 12117 0.043284893689986
20 15267 0.054537465706447
21 18327 0.065468535665295
22 20993 0.074992141060814
23 22967 0.082043752857796
24 24017 0.085794610196616
25 24017 0.085794610196616
26 22967 0.082043752857796
27 20993 0.074992141060814
28 18327 0.065468535665295
29 15267 0.054537465706447
30 12117 0.043284893689986
31 9142 0.032657464563329
32 6538 0.023355338363054
33 4417 0.015778606538637
34 2807 0.010027291952446
35 1667 0.005954932556013
36 917 0.003275748742570
37 462 0.001650377229081
38 210 0.000750171467764
39 84 0.000300068587106
40 28 0.000100022862369
41 7 0.000025005715592
42 1 0.000003572245085
Total 279936 1.000000000000000

8 Dice

8 Dice

Total Combinations Probability
8 1 0.000000595374181
9 8 0.000004762993446
10 36 0.000021433470508
11 120 0.000071444901692
12 330 0.000196473479652
13 792 0.000471536351166
14 1708 0.001016899100747
15 3368 0.002005220240817
16 6147 0.003659765089163
17 10480 0.006239521414419
18 16808 0.010007049230300
19 25488 0.015174897119342
20 36688 0.021843087943911
21 50288 0.029940176802317
22 65808 0.039180384087792
23 82384 0.049049306508154
24 98813 0.058830708923945
25 113688 0.067686899862826
26 125588 0.074771852613931
27 133288 0.079356233805822
28 135954 0.080943501371742
29 133288 0.079356233805822
30 125588 0.074771852613931
31 113688 0.067686899862826
32 98813 0.058830708923945
33 82384 0.049049306508154
34 65808 0.039180384087792
35 50288 0.029940176802317
36 36688 0.021843087943911
37 25488 0.015174897119342
38 16808 0.010007049230300
39 10480 0.006239521414419
40 6147 0.003659765089163
41 3368 0.002005220240817
42 1708 0.001016899100747
43 792 0.000471536351166
44 330 0.000196473479652
45 120 0.000071444901692
46 36 0.000021433470508
47 8 0.000004762993446
48 1 0.000000595374181
Total 1679616 1.000000000000000

9 Dice

9 Dice

Total Combinations Probability
9 1 0.000000099229030
10 9 0.000000893061271
11 45 0.000004465306356
12 165 0.000016372789971
13 495 0.000049118369913
14 1287 0.000127707761774
15 2994 0.000297091716202
16 6354 0.000630501257430
17 12465 0.001236889860540
18 22825 0.002264902612661
19 39303 0.003899998571102
20 63999 0.006350558699131
21 98979 0.009821590172992
22 145899 0.014477416266575
23 205560 0.020397519433013
24 277464 0.027532483615303
25 359469 0.035669760230910
26 447669 0.044421760688157
27 536569 0.053243221466494
28 619569 0.061479230967078
29 689715 0.068439750514403
30 740619 0.073490905064015
31 767394 0.076147762345679
32 767394 0.076147762345679
33 740619 0.073490905064015
34 689715 0.068439750514403
35 619569 0.061479230967078
36 536569 0.053243221466494
37 447669 0.044421760688157
38 359469 0.035669760230910
39 277464 0.027532483615303
40 205560 0.020397519433013
41 145899 0.014477416266575
42 98979 0.009821590172992
43 63999 0.006350558699131
44 39303 0.003899998571102
45 22825 0.002264902612661
46 12465 0.001236889860540
47 6354 0.000630501257430
48 2994 0.000297091716202
49 1287 0.000127707761774
50 495 0.000049118369913
51 165 0.000016372789971
52 45 0.000004465306356
53 9 0.000000893061271
54 1 0.000000099229030
Total 10077696 1.000000000000000

10 Dice

10 Dice

Total Combinations Probability
10 1 0.000000016538172
11 10 0.000000165381717
12 55 0.000000909599443
13 220 0.000003638397771
14 715 0.000011824792757
15 2002 0.000033109419719
16 4995 0.000082608167581
17 11340 0.000187542866941
18 23760 0.000392946959305
19 46420 0.000767701929753
20 85228 0.001409515296618
21 147940 0.002446657119511
22 243925 0.004034073528976
23 383470 0.006341892697167
24 576565 0.009535330959246
25 831204 0.013746594459686
26 1151370 0.019041554736321
27 1535040 0.025386755067825
28 1972630 0.032623693616742
29 2446300 0.040457329400159
30 2930455 0.048464367913724
31 3393610 0.056124104821843
32 3801535 0.062870438507638
33 4121260 0.068158105450558
34 4325310 0.071532719383478
35 4395456 0.072692805974699
36 4325310 0.071532719383478
37 4121260 0.068158105450558
38 3801535 0.062870438507638
39 3393610 0.056124104821843
40 2930455 0.048464367913724
41 2446300 0.040457329400159
42 1972630 0.032623693616742
43 1535040 0.025386755067825
44 1151370 0.019041554736321
45 831204 0.013746594459686
46 576565 0.009535330959246
47 383470 0.006341892697167
48 243925 0.004034073528976
49 147940 0.002446657119511
50 85228 0.001409515296618
51 46420 0.000767701929753
52 23760 0.000392946959305
53 11340 0.000187542866941
54 4995 0.000082608167581
55 2002 0.000033109419719
56 715 0.000011824792757
57 220 0.000003638397771
58 55 0.000000909599443
59 10 0.000000165381717
60 1 0.000000016538172
Total 60466176 1.000000000000000

15 Dice

15 Dice

Total Combinations Probability
15 1 0.000000000002127
16 15 0.000000000031902
17 120 0.000000000255219
18 680 0.000000001446239
19 3060 0.000000006508077
20 11628 0.000000024730692
21 38745 0.000000082403737
22 116055 0.000000246828384
23 317970 0.000000676265747
24 806990 0.000001716324482
25 1915356 0.000004073622219
26 4282980 0.000009109138191
27 9076405 0.000019303902289
28 18315675 0.000038954189523
29 35332650 0.000075146274677
30 65372310 0.000139035299179
31 116325135 0.000247402913355
32 199542465 0.000424391402418
33 330639445 0.000703211407949
34 530137275 0.001127507879645
35 823747890 0.001751965539144
36 1242073550 0.002641669961281
37 1819496655 0.003869746407663
38 2592085185 0.005512905069347
39 3594444165 0.007644744691797
40 4855600971 0.010327001351136
41 6394206690 0.013599342598671
42 8213538270 0.017468737921112
43 10296957375 0.021899800531243
44 12604578705 0.026807701475977
45 15071885925 0.032055225963014
46 17610885675 0.037455227735273
47 20114111295 0.042779144283262
48 22461407505 0.047771426655096
49 24529001175 0.052168831374144
50 26199964377 0.055722673493341
51 27374880105 0.058221510688364
52 27981391815 0.059511453434084
53 27981391815 0.059511453434084
54 27374880105 0.058221510688364
55 26199964377 0.055722673493341
56 24529001175 0.052168831374144
57 22461407505 0.047771426655096
58 20114111295 0.042779144283262
59 17610885675 0.037455227735273
60 15071885925 0.032055225963014
61 12604578705 0.026807701475977
62 10296957375 0.021899800531243
63 8213538270 0.017468737921112
64 6394206690 0.013599342598671
65 4855600971 0.010327001351136
66 3594444165 0.007644744691797
67 2592085185 0.005512905069347
68 1819496655 0.003869746407663
69 1242073550 0.002641669961281
70 823747890 0.001751965539144
71 530137275 0.001127507879645
72 330639445 0.000703211407949
73 199542465 0.000424391402418
74 116325135 0.000247402913355
75 65372310 0.000139035299179
76 35332650 0.000075146274677
77 18315675 0.000038954189523
78 9076405 0.000019303902289
79 4282980 0.000009109138191
80 1915356 0.000004073622219
81 806990 0.000001716324482
82 317970 0.000000676265747
83 116055 0.000000246828384
84 38745 0.000000082403737
85 11628 0.000000024730692
86 3060 0.000000006508077
87 680 0.000000001446239
88 120 0.000000000255219
89 15 0.000000000031902
90 1 0.000000000002127
Total 470184984576 1.000000000000000

20 Dice

20 Dice

Total Combinations Probability
20 1 0.000000000000000
21 20 0.000000000000005
22 210 0.000000000000057
23 1540 0.000000000000421
24 8855 0.000000000002422
25 42504 0.000000000011625
26 177080 0.000000000048433
27 657400 0.000000000179806
28 2215875 0.000000000606066
29 6876100 0.000000001880690
30 19852910 0.000000005429992
31 53777220 0.000000014708668
32 137578715 0.000000037629309
33 334221400 0.000000091413270
34 774447600 0.000000211820033
35 1718122120 0.000000469925510
36 3661011200 0.000001001327284
37 7513026360 0.000002054896275
38 14883869260 0.000004070903793
39 28522751000 0.000007801289651
40 52968655260 0.000014487516372
41 95473613400 0.000026113095197
42 167259698400 0.000045747387905
43 285156111240 0.000077993368153
44 473630804445 0.000129543293107
45 767177392956 0.000209831550118
46 1212940768770 0.000331752791531
47 1873365928060 0.000512386418360
48 2828541558365 0.000773637577456
49 4177796844360 0.001142673905644
50 6040016957080 0.001652011819536
51 8552094731000 0.002339093031989
52 11864959484010 0.003245198390200
53 16136741780240 0.004413578362310
54 21522858933280 0.005886741312258
55 28163138907504 0.007702931743576
56 36166519159030 0.009891945262199
57 45594319698440 0.012470553573071
58 56443534859400 0.015437934593017
59 68631941843000 0.018771599471991
60 81987009993775 0.022424359156701
61 96240548343540 0.026322860434320
62 111030711156790 0.030368134471458
63 125912390300660 0.034438439242937
64 140376201624375 0.038394452517752
65 153875306132440 0.042086607748265
66 165858316337600 0.045364094323752
67 175805662952520 0.048084804265072
68 183266172913710 0.050125336721062
69 187890345960720 0.051390099483075
70 189456975899496 0.051818590196609
71 187890345960720 0.051390099483075
72 183266172913710 0.050125336721062
73 175805662952520 0.048084804265072
74 165858316337600 0.045364094323752
75 153875306132440 0.042086607748265
76 140376201624375 0.038394452517752
77 125912390300660 0.034438439242937
78 111030711156790 0.030368134471458
79 96240548343540 0.026322860434320
80 81987009993775 0.022424359156701
81 68631941843000 0.018771599471991
82 56443534859400 0.015437934593017
83 45594319698440 0.012470553573071
84 36166519159030 0.009891945262199
85 28163138907504 0.007702931743576
86 21522858933280 0.005886741312258
87 16136741780240 0.004413578362310
88 11864959484010 0.003245198390200
89 8552094731000 0.002339093031989
90 6040016957080 0.001652011819536
91 4177796844360 0.001142673905644
92 2828541558365 0.000773637577456
93 1873365928060 0.000512386418360
94 1212940768770 0.000331752791531
95 767177392956 0.000209831550118
96 473630804445 0.000129543293107
97 285156111240 0.000077993368153
98 167259698400 0.000045747387905
99 95473613400 0.000026113095197
100 52968655260 0.000014487516372
101 28522751000 0.000007801289651
102 14883869260 0.000004070903793
103 7513026360 0.000002054896275
104 3661011200 0.000001001327284
105 1718122120 0.000000469925510
106 774447600 0.000000211820033
107 334221400 0.000000091413270
108 137578715 0.000000037629309
109 53777220 0.000000014708668
110 19852910 0.000000005429992
111 6876100 0.000000001880690
112 2215875 0.000000000606066
113 657400 0.000000000179806
114 177080 0.000000000048433
115 42504 0.000000000011625
116 8855 0.000000000002422
117 1540 0.000000000000421
118 210 0.000000000000057
119 20 0.000000000000005
120 1 0.000000000000000
Total 3656158440062970 1.000000000000000

25 Dice

25 Dice

Total Combinations Probability
25 1 0.000000000000000
26 25 0.000000000000000
27 325 0.000000000000000
28 2925 0.000000000000000
29 20475 0.000000000000001
30 118755 0.000000000000004
31 593750 0.000000000000021
32 2628950 0.000000000000092
33 10510175 0.000000000000370
34 38493975 0.000000000001354
35 130616265 0.000000000004594
36 414257025 0.000000000014571
37 1236833625 0.000000000043504
38 3496735425 0.000000000122993
39 9406694100 0.000000000330869
40 24177540660 0.000000000850415
41 59580040650 0.000000002095654
42 141189459450 0.000000004966164
43 322583308750 0.000000011346466
44 712199564350 0.000000025050733
45 1522455240070 0.000000053550468
46 3156698136150 0.000000111032928
47 6358359922200 0.000000223647397
48 12459084463800 0.000000438232791
49 23779313297000 0.000000836407752
50 44256184906376 0.000001556656227
51 80399354535800 0.000002827947239
52 142704104517400 0.000005019439281
53 247679998965100 0.000008711835726
54 420677492766700 0.000014796807276
55 699707258309580 0.000024611332026
56 1140438281829900 0.000040113497290
57 1822520072208520 0.000064104875416
58 2857290162402720 0.000100501625569
59 4396790346715320 0.000154651628646
60 6643817535172800 0.000233688013578
61 9862478344967270 0.000346900401944
62 14388411163369000 0.000506094456314
63 20637508476323300 0.000725898677305
64 29111652608272900 0.001023966151086
65 40399719541288500 0.001421009857472
66 55171967687560400 0.001940605301975
67 74165974148054000 0.002608695841209
68 98162566957232600 0.003452746129547
69 127950767866365000 0.004500509025152
70 164281617348912000 0.005778401442010
71 207811873678121000 0.007309524035048
72 259039887536427000 0.009111405669324
73 318237332924750000 0.011193602139723
74 385381765170843000 0.013555323982829
75 460095994386015000 0.016183304013721
76 541600822637078000 0.019050134914810
77 628687635496274000 0.022113305177719
78 719716557311938000 0.025315134217222
79 812644347034221000 0.028583753572202
80 905083995793710000 0.031835203176306
81 994395256747846000 0.034976615632631
82 1077802349727290000 0.037910356328175
83 1152532159899090000 0.040538884399827
84 1215963739043970000 0.042770011255738
85 1265778148181130000 0.044522171103177
86 1300096912039730000 0.045729290912251
87 1317597742043220000 0.046344860827866
88 1317597742043220000 0.046344860827866
89 1300096912039730000 0.045729290912251
90 1265778148181130000 0.044522171103177
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Written by: Michael Shackleford

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