Last updated: Oct 16, 2023
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
91
1215963739043970000
0.042770011255738
92
1152532159899090000
0.040538884399827
93
1077802349727290000
0.037910356328175
94
994395256747846000
0.034976615632631
95
905083995793710000
0.031835203176306
96
812644347034221000
0.028583753572202
97
719716557311938000
0.025315134217222
98
628687635496274000
0.022113305177719
99
541600822637078000
0.019050134914810
100
460095994386015000
0.016183304013721
101
385381765170843000
0.013555323982829
102
318237332924750000
0.011193602139723
103
259039887536427000
0.009111405669324
104
207811873678121000
0.007309524035048
105
164281617348912000
0.005778401442010
106
127950767866365000
0.004500509025152
107
98162566957232600
0.003452746129547
108
74165974148054000
0.002608695841209
109
55171967687560400
0.001940605301975
110
40399719541288500
0.001421009857472
111
29111652608272900
0.001023966151086
112
20637508476323300
0.000725898677305
113
14388411163369000
0.000506094456314
114
9862478344967270
0.000346900401944
115
6643817535172800
0.000233688013578
116
4396790346715320
0.000154651628646
117
2857290162402720
0.000100501625569
118
1822520072208520
0.000064104875416
119
1140438281829900
0.000040113497290
120
699707258309580
0.000024611332026
121
420677492766700
0.000014796807276
122
247679998965100
0.000008711835726
123
142704104517400
0.000005019439281
124
80399354535800
0.000002827947239
125
44256184906376
0.000001556656227
126
23779313297000
0.000000836407752
127
12459084463800
0.000000438232791
128
6358359922200
0.000000223647397
129
3156698136150
0.000000111032928
130
1522455240070
0.000000053550468
131
712199564350
0.000000025050733
132
322583308750
0.000000011346466
133
141189459450
0.000000004966164
134
59580040650
0.000000002095654
135
24177540660
0.000000000850415
136
9406694100
0.000000000330869
137
3496735425
0.000000000122993
138
1236833625
0.000000000043504
139
414257025
0.000000000014571
140
130616265
0.000000000004594
141
38493975
0.000000000001354
142
10510175
0.000000000000370
143
2628950
0.000000000000092
144
593750
0.000000000000021
145
118755
0.000000000000004
146
20475
0.000000000000001
147
2925
0.000000000000000
148
325
0.000000000000000
149
25
0.000000000000000
150
1
0.000000000000000
Written by: Michael Shackleford
