Factorial Calculator

Q: How many digits does 100! have?

100! has exactly 158 digits. This is computed as ⌊log₁₀(100!)⌋ + 1 = ⌊∑ log₁₀(k) for k=1 to 100⌋ + 1 = 157 + 1 = 158.

Q: In how many ways can a deck of 52 cards be shuffled?

Exactly 52! ≈ 8.07 × 1067 ways. This number is so astronomically large that every shuffle performed in all of human history is almost certainly unique.

Q: What is the largest factorial this tool can compute exactly?

This tool computes exact integer values up to 170! — the largest factorial that fits in an IEEE 754 double. 170! has 307 digits.

Q: What is the connection between factorials and the Gamma function?

The Gamma function Γ(z) generalises factorials to complex numbers: Γ(n+1) = n! for any non-negative integer n. Γ(1/2) = √π, and Γ(3/2) = √π/2.

Enter any integer from 0 to 170 to get the exact value, number of digits, scientific notation, and Stirling approximation. Integers from 0 to 170.

Integers from 0 to 170 (171! overflows IEEE 754 double).

Most calculated factorials

10! calculated 51×
5! calculated 25×
100! calculated 18×
52! calculated 17×
0! calculated 12×
101! calculated 8×
25! calculated 8×
1! calculated 5×

Click any number to see its full factorial value.

Factorial table: 0! to 170!

Exact values for 0!–20!, scientific notation for 21!–170!. Each row multiplies the previous factorial by n.

n	n!	Digits
0	1	1
1	1	1
2	2	1
3	6	1
4	24	2
5	120	3
6	720	3
7	5,040	4
8	40,320	5
9	362,880	6
10	3,628,800	7
11	39,916,800	8
12	479,001,600	9
13	6,227,020,800	10
14	87,178,291,200	11
15	1,307,674,368,000	13
16	20,922,789,888,000	14
17	355,687,428,096,000	15
18	6,402,373,705,728,000	16
19	121,645,100,408,832,000	18
20	2,432,902,008,176,640,000	19

Show 21! – 170!

n	n!	Digits
21	5.1090942 × 10¹⁹	20
22	1.1240007 × 10²¹	22
23	2.5852016 × 10²²	23
24	6.2044840 × 10²³	24
25	1.5511210 × 10²⁵	26
26	4.0329146 × 10²⁶	27
27	1.0888869 × 10²⁸	29
28	3.0488834 × 10²⁹	30
29	8.8417619 × 10³⁰	31
30	2.6525285 × 10³²	33
31	8.2228386 × 10³³	34
32	2.6313083 × 10³⁵	36
33	8.6833176 × 10³⁶	37
34	2.9523279 × 10³⁸	39
35	1.0333147 × 10⁴⁰	41
36	3.7199332 × 10⁴¹	42
37	1.3763753 × 10⁴³	44
38	5.2302261 × 10⁴⁴	45
39	2.0397882 × 10⁴⁶	47
40	8.1591528 × 10⁴⁷	48
41	3.3452526 × 10⁴⁹	50
42	1.4050061 × 10⁵¹	52
43	6.0415263 × 10⁵²	53
44	2.6582715 × 10⁵⁴	55
45	1.1962222 × 10⁵⁶	57
46	5.5026221 × 10⁵⁷	58
47	2.5862324 × 10⁵⁹	60
48	1.2413915 × 10⁶¹	62
49	6.0828186 × 10⁶²	63
50	3.0414093 × 10⁶⁴	65
51	1.5511187 × 10⁶⁶	67
52	8.0658175 × 10⁶⁷	68
53	4.2748832 × 10⁶⁹	70
54	2.3084369 × 10⁷¹	72
55	1.2696403 × 10⁷³	74
56	7.1099858 × 10⁷⁴	75
57	4.0526919 × 10⁷⁶	77
58	2.3505613 × 10⁷⁸	79
59	1.3868311 × 10⁸⁰	81
60	8.3209871 × 10⁸¹	82
61	5.0758021 × 10⁸³	84
62	3.1469973 × 10⁸⁵	86
63	1.9826083 × 10⁸⁷	88
64	1.2688693 × 10⁸⁹	90
65	8.2476505 × 10⁹⁰	91
66	5.4434493 × 10⁹²	93
67	3.6471110 × 10⁹⁴	95
68	2.4800355 × 10⁹⁶	97
69	1.7112245 × 10⁹⁸	99
70	1.1978571 × 10¹⁰⁰	101
71	8.5047858 × 10¹⁰¹	102
72	6.1234458 × 10¹⁰³	104
73	4.4701154 × 10¹⁰⁵	106
74	3.3078854 × 10¹⁰⁷	108
75	2.4809140 × 10¹⁰⁹	110
76	1.8854947 × 10¹¹¹	112
77	1.4518309 × 10¹¹³	114
78	1.1324281 × 10¹¹⁵	116
79	8.9461821 × 10¹¹⁶	117
80	7.1569457 × 10¹¹⁸	119
81	5.7971260 × 10¹²⁰	121
82	4.7536433 × 10¹²²	123
83	3.9455239 × 10¹²⁴	125
84	3.3142401 × 10¹²⁶	127
85	2.8171041 × 10¹²⁸	129
86	2.4227095 × 10¹³⁰	131
87	2.1077572 × 10¹³²	133
88	1.8548264 × 10¹³⁴	135
89	1.6507955 × 10¹³⁶	137
90	1.4857159 × 10¹³⁸	139
91	1.3520015 × 10¹⁴⁰	141
92	1.2438414 × 10¹⁴²	143
93	1.1567725 × 10¹⁴⁴	145
94	1.0873661 × 10¹⁴⁶	147
95	1.0329978 × 10¹⁴⁸	149
96	9.9167793 × 10¹⁴⁹	150
97	9.6192759 × 10¹⁵¹	152
98	9.4268904 × 10¹⁵³	154
99	9.3326215 × 10¹⁵⁵	156
100	9.3326215 × 10¹⁵⁷	158
101	9.4259477 × 10¹⁵⁹	160
102	9.6144667 × 10¹⁶¹	162
103	9.9029007 × 10¹⁶³	164
104	1.0299016 × 10¹⁶⁶	167
105	1.0813967 × 10¹⁶⁸	169
106	1.1462805 × 10¹⁷⁰	171
107	1.2265202 × 10¹⁷²	173
108	1.3246418 × 10¹⁷⁴	175
109	1.4438595 × 10¹⁷⁶	177
110	1.5882455 × 10¹⁷⁸	179
111	1.7629525 × 10¹⁸⁰	181
112	1.9745068 × 10¹⁸²	183
113	2.2311927 × 10¹⁸⁴	185
114	2.5435597 × 10¹⁸⁶	187
115	2.9250936 × 10¹⁸⁸	189
116	3.3931086 × 10¹⁹⁰	191
117	3.9699371 × 10¹⁹²	193
118	4.6845258 × 10¹⁹⁴	195
119	5.5745857 × 10¹⁹⁶	197
120	6.6895029 × 10¹⁹⁸	199
121	8.0942985 × 10²⁰⁰	201
122	9.8750442 × 10²⁰²	203
123	1.2146304 × 10²⁰⁵	206
124	1.5061417 × 10²⁰⁷	208
125	1.8826771 × 10²⁰⁹	210
126	2.3721732 × 10²¹¹	212
127	3.0126600 × 10²¹³	214
128	3.8562048 × 10²¹⁵	216
129	4.9745042 × 10²¹⁷	218
130	6.4668554 × 10²¹⁹	220
131	8.4715806 × 10²²¹	222
132	1.1182486 × 10²²⁴	225
133	1.4872707 × 10²²⁶	227
134	1.9929427 × 10²²⁸	229
135	2.6904727 × 10²³⁰	231
136	3.6590428 × 10²³²	233
137	5.0128887 × 10²³⁴	235
138	6.9177864 × 10²³⁶	237
139	9.6157231 × 10²³⁸	239
140	1.3462012 × 10²⁴¹	242
141	1.8981437 × 10²⁴³	244
142	2.6953641 × 10²⁴⁵	246
143	3.8543707 × 10²⁴⁷	248
144	5.5502938 × 10²⁴⁹	250
145	8.0479260 × 10²⁵¹	252
146	1.1749972 × 10²⁵⁴	255
147	1.7272458 × 10²⁵⁶	257
148	2.5563239 × 10²⁵⁸	259
149	3.8089226 × 10²⁶⁰	261
150	5.7133839 × 10²⁶²	263
151	8.6272097 × 10²⁶⁴	265
152	1.3113358 × 10²⁶⁷	268
153	2.0063439 × 10²⁶⁹	270
154	3.0897696 × 10²⁷¹	272
155	4.7891429 × 10²⁷³	274
156	7.4710629 × 10²⁷⁵	276
157	1.1729568 × 10²⁷⁸	279
158	1.8532718 × 10²⁸⁰	281
159	2.9467022 × 10²⁸²	283
160	4.7147236 × 10²⁸⁴	285
161	7.5907050 × 10²⁸⁶	287
162	1.2296942 × 10²⁸⁹	290
163	2.0044015 × 10²⁹¹	292
164	3.2872185 × 10²⁹³	294
165	5.4239106 × 10²⁹⁵	296
166	9.0036917 × 10²⁹⁷	298
167	1.5036165 × 10³⁰⁰	301
168	2.5260757 × 10³⁰²	303
169	4.2690680 × 10³⁰⁴	305
170	7.2574156 × 10³⁰⁶	307

About factorials

What is a factorial?

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1 (the empty product).

How fast do factorials grow?

Factorials grow faster than any exponential function. While 2ⁿ doubles at each step, n! multiplies by n — an ever-increasing factor. By n = 100, we already have 100! ≈ 9.33 × 10¹⁵⁷, and 170! reaches about 7.26 × 10³⁰⁶.

What is Stirling's approximation?

Stirling's formula provides a practical way to estimate large factorials: n! ≈ √(2πn) · (n/e)ⁿ. The relative error is already below 1% for n = 10 and below 0.1% for n = 100. It is widely used in combinatorics, statistical mechanics, and information theory.

Factorials in combinatorics

Factorials count permutations: n! is the number of ways to arrange n distinct objects in a row. They also appear in combinations C(n,k) = n! / (k!(n−k)!), in Taylor series (the n-th term is divided by n!), and in the Gamma function: Γ(n+1) = n!.

Why does 171! overflow?

IEEE 754 double-precision floats can represent values up to ≈ 1.8 × 10³⁰⁸. Since 170! ≈ 7.26 × 10³⁰⁶ is within range but 171! ≈ 1.24 × 10³⁰⁹ is not, 171! evaluates to Infinity in most languages using native floats.

Frequently asked questions

What is 0! (zero factorial)?

0! = 1 by convention. This follows from the definition of the empty product: the product of no numbers is the multiplicative identity, 1. It also ensures that C(n,0) = n!/(0!·n!) = 1 remains consistent for all n.

How many digits does 100! have?

100! has exactly 158 digits. This is computed as ⌊log₁₀(100!)⌋ + 1 = ⌊∑ log₁₀(k) for k=1 to 100⌋ + 1 = 157 + 1 = 158.

In how many ways can a deck of 52 cards be shuffled?

Exactly 52! ≈ 8.07 × 10⁶⁷ ways. This number is so astronomically large that every shuffle performed in all of human history is almost certainly unique.

What is the largest factorial this tool can compute exactly?

This tool computes exact integer values up to 170! — the largest factorial that fits in an IEEE 754 double. 170! has 307 digits.

How is the number of digits of n! calculated without computing n!?

The digit count is ⌊log₁₀(n!)⌋ + 1. By logarithm properties, log₁₀(n!) = ∑ log₁₀(k) for k=1 to n. This is computable in O(n) without big-integer arithmetic.

What is the connection between factorials and the Gamma function?

The Gamma function Γ(z) generalises factorials to complex numbers: Γ(n+1) = n! for any non-negative integer n. Γ(1/2) = √π, and Γ(3/2) = √π/2.