How many unique bingo cards can a word list make?
Short answer: for a 5 x 5 card with a free centre, a list of n words can make C(n, 24) cards with different sets of words: 1 card from 24 words, 25 from 25 words, 325 from 26, 593,775 from 30 and about 122 trillion from 50. Different arrangements are never the limit; a single set of 24 words can be laid out in 24! ways, which is about 620 sextillion. If your list has 26 or more words (27 without the free centre) you can print 200 cards that are all different in both senses, and 48 or more makes them feel different too.
What "unique" means
Two bingo cards can differ in two ways, and it matters which one you care about.
- Different arrangement. The same words in a different layout. Every card the generator prints has a different arrangement from every other card in the game, always. For a line game this is what matters: two cards with the same words but different layouts complete lines on different calls.
- Different set. A different selection of words. Two cards with the same set of 24 words will both fill up on exactly the same call, so in a full-card ("blackout") game they tie. The generator also guarantees different sets whenever the list is long enough to allow it, and warns you when it isn't.
People asking "can every card be unique?" usually mean the second kind, and the answer depends only on how many words are in the list and how many squares are on the card.
How many squares a card has
A 5 x 5 card has 25 squares, or 24 words plus a free centre. A 4 x 4 card has 16 squares and no centre, so no free space. A 3 x 3 card has 9 squares, or 8 words plus a free centre. The number of words a card holds, call it k, is 24, 25, 16, 9 or 8 depending on the grid and the free space.
The counting
A card's set is a choice of k words from your list of n, and the number of ways to choose k things from n without caring about order is the binomial coefficient:
C(n, k) = n! ÷ (k! × (n − k)!)
For a 5 x 5 card with a free centre and a list of 30 words that is C(30, 24) = 30! ÷ (24! × 6!) = 593,775 different sets. Each set can be arranged on the grid in 24! = 620,448,401,733,239,439,360,000 ways, so the number of different cards is 593,775 × 24!, an absurd figure. Arrangements are effectively unlimited; the sets are what run out.
Why 24 words give every card the same set
If your list has exactly 24 words and the card has 24 squares, there is only one way to choose 24 from 24: C(24, 24) = 1. Every card holds the whole list. The cards still look different because the words are in different places, so line games work normally, but in a full-card game every player finishes on the same call, the 24th. It is also a dull game, since every word called is on every card.
With 25 words, each card leaves out exactly one word, so there are 25 possible sets and a game of up to 25 cards can have them all different. Beyond 25 cards, sets must repeat. With 26 words the count jumps to 325, and from there it grows fast, as the table shows.
Table: different sets by list length
| Words in list | 5 x 5, free centre (24) | 5 x 5, no free space (25) | 4 x 4 (16) |
|---|---|---|---|
| 24 | 1 | 0 | 735,471 |
| 25 | 25 | 1 | 2,042,975 |
| 26 | 325 | 26 | 5,311,735 |
| 27 | 2,925 | 351 | 13,037,895 |
| 28 | 20,475 | 3,276 | 30,421,755 |
| 29 | 118,755 | 23,751 | 67,863,915 |
| 30 | 593,775 | 142,506 | 145,422,675 |
| 31 | 2,629,575 | 736,281 | 300,540,195 |
| 32 | 10,518,300 | 3,365,856 | 601,080,390 |
| 33 | 38,567,100 | 13,884,156 | 1,166,803,110 |
| 34 | 131,128,140 | 52,451,256 | 2,203,961,430 |
| 35 | 417,225,900 | 183,579,396 | 4,059,928,950 |
| 36 | 1,251,677,700 | 600,805,296 | 7,307,872,110 |
| 37 | 3,562,467,300 | 1,852,482,996 | 12,875,774,670 |
| 38 | 9,669,554,100 | 5,414,950,296 | 22,239,974,430 |
| 39 | 25,140,840,660 | 15,084,504,396 | 37,711,260,990 |
| 40 | 62,852,101,650 | 40,225,345,056 | 62,852,101,650 |
| 41 | 151,584,480,450 | 103,077,446,706 | 103,077,446,706 |
| 42 | 353,697,121,050 | 254,661,927,156 | 166,509,721,602 |
| 43 | 800,472,431,850 | 608,359,048,206 | 265,182,149,218 |
| 44 | 1,761,039,350,070 | 1,408,831,480,056 | 416,714,805,914 |
| 45 | 3,773,655,750,150 | 3,169,870,830,126 | 646,626,422,970 |
| 46 | 7,890,371,113,950 | 6,943,526,580,276 | 991,493,848,554 |
| 47 | 16,123,801,841,550 | 14,833,897,694,226 | 1,503,232,609,098 |
| 48 | 32,247,603,683,100 | 30,957,699,535,776 | 2,254,848,913,647 |
| 49 | 63,205,303,218,876 | 63,205,303,218,876 | 3,348,108,992,991 |
| 50 | 121,548,660,036,300 | 126,410,606,437,752 | 4,923,689,695,575 |
| 51 | 229,591,913,401,900 | 247,959,266,474,052 | 7,174,519,270,695 |
| 52 | 426,384,982,032,100 | 477,551,179,875,952 | 10,363,194,502,115 |
| 53 | 779,255,311,989,700 | 903,936,161,908,052 | 14,844,575,908,435 |
| 54 | 1,402,659,561,581,460 | 1,683,191,473,897,752 | 21,094,923,659,355 |
| 55 | 2,488,589,544,741,300 | 3,085,851,035,479,212 | 29,749,251,314,475 |
| 56 | 4,355,031,703,297,275 | 5,574,440,580,220,512 | 41,648,951,840,265 |
| 57 | 7,522,327,487,513,475 | 9,929,472,283,517,787 | 57,902,201,338,905 |
| 58 | 12,832,205,713,993,575 | 17,451,799,771,031,262 | 79,960,182,801,345 |
| 59 | 21,631,432,489,303,455 | 30,284,005,485,024,837 | 109,712,808,959,985 |
| 60 | 36,052,387,482,172,425 | 51,915,437,974,328,292 | 149,608,375,854,525 |
A 3 x 3 card with a free centre holds 8 words; C(n, 8) is 1 for 8 words, 9 for 9 words, 45 for 10, 495 for 12 and 125,970 for 20, so 16 words is already plenty for a preschool class.
Enough sets is not the same as enough variety
The table says 26 words can make 325 different cards. It doesn't say they will feel different. Two cards drawn at random from a list of n words share, on average, 24 × 24 ÷ n squares. From 26 words two cards share about 22.2 of their 24 squares; from 30 words, 19.2; from 40 words, 14.4; from 50, 11.5; from 75, 7.7. A game drawn from 30 words has cards that are mostly the same, so lines fall for several players within a call or two of each other and the room shouts bingo at once. That is why the practical advice is 48 to 75 words even though 26 is mathematically enough for 200 cards, and why the generator shows a warning below twice the squares on a card.
For comparison: classic 75-ball cards
A standard number card draws each column from its own range of 15 numbers (B 1 to 15, I 16 to 30, N 31 to 45, G 46 to 60, O 61 to 75), five per column except the N column, which has four around the free space. That is C(15, 5)4 × C(15, 4) = 111,007,923,832,370,565 different sets, about 1.1 × 1017, which is why nobody has ever worried about two number cards matching. A 24-word list with the structure of ordinary free-choice cards is a much smaller space, which is the whole reason this page exists.
What the generator does with this
- It always rejects a card whose arrangement duplicates an earlier card, so no two players ever hold the same layout.
- It also repairs a card whose set duplicates an earlier card's, as long as C(n, k) is at least the number of cards requested. If it isn't, the game is still made, but a warning says so. Rather than drawing cards at random until it finds enough distinct sets, which could take a long time near the limit, it deals words from a shuffled bag so every word is used about equally often, and swaps one word whenever a set repeats; the methodology page has the details.
- The caller sheet lists every word in the list once, in a seeded order, so the printed sheet and the on-screen caller always agree. Because cards are dealt evenly, a class-sized game uses every word.