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## List of numbers appearing in sports-related combinatorics

• Some association football competitions, such as the UEFA Cup, have five-team single round-robin tournaments in the group stage. With three points for a win, there are exactly 355 possible points columns in the final standings of a group.
• The number π is approximately equal to 355/113.
• And some years in the Hebrew and Islamic calendars have 355 days.
• Some association football competitions, such as the UEFA Champions League, have four-team double round-robin tournaments in the group stage. With three points for a win, there are exactly 748 possible points columns in the final standings of a group.
• There are 8!/4! = 14!!!! = 1,680 possible ways to draw the quarter-finals of a single-elimination tournament.
• In the FIFA World Cup qualification intercontinental play-offs, one team each of AFC, CONCACAF, CONMEBOL and OFC plays another. The number of possible matches is currently equal to either 3,421 (ignoring home advantage) or 6,842 (recognizing home advantage).
• In the FIFA World Cup qualification, teams of the same confederation play another. The number of possible matches is currently equal to either 4,646 (ignoring home advantage) or 9,292 (recognizing home advantage).
• For the round of 16 of the UEFA Champions League, the eight group winners have to be drawn against a runner-up of another group. The number of possible outcomes is !8 = 14,833. In reality, the actual number is almost always smaller, since teams of the same association cannot be drawn against each other.
• In the group stage of the FIFA World Cup, teams of different confederations, or sometimes two UEFA teams, play another. The number of possible matches is currently equal to 18,994.
• Since there are 211 FIFA members, and at most one of the 11 OFC members may qualify for the FIFA World Cup, there are 211C211C2 = 22,100 possible final matches.
• Ignoring chronological order, there are 10!/5! = 18!!!! = 30,240 possible combinations for a matchday in a 10-team league, such as the CONMEBOL group of the FIFA World Cup qualification.
• The marathon goes over 42,195 metres or 42.195 km or 26.22 miles. It is equal to 291C2.
• Since there are 211 FIFA members, and at most one of the 11 OFC members may qualify for the FIFA World Cup, there are 211P211P2 = 44,200 possible combinations for the two highest ranked nations.
• In some countries, there are football pools involving 11 matches. The number of possible combinations is 311 = 177,147.
• In some countries, there are football pools involving 12 matches. The number of possible combinations is 312 = 531,441.
• For each match of certain international football tournaments, such as the FIFA World Cup, each national team has to choose one goalkeeper out of three, and 10 outfield players out of 20. The number of possible combinations is 3 × 20C10 = 554,268.
• Ignoring chronological order, there are 12!/6! = 22!!!! = 665,280 possible combinations for a matchday in a 12-team league, such as the Scottish Premiership.
• In some countries, there are football pools involving 13 matches. The number of possible combinations is 313 = 1,594,323.
• In some countries, there are football pools involving 14 matches. The number of possible combinations is 314 = 4,782,969.
• Since there are 211 FIFA members, and at most one of the 11 OFC members may qualify for the FIFA World Cup, there are 200P3 + 3 × 11 × 200P2 = 9,193,800 possible combinations for the three highest ranked nations.
• For the round of 16 of the UEFA Women's Champions League, the eight seeded teams have to be drawn against the eight unseeded teams. The number of possible outcomes is 8! × 28 = 16!! = 10,321,920. In reality, the actual number is almost always smaller, since teams of the same association cannot be drawn against each other.
• Ignoring chronological order, there are 14!/7! = 26!!!! = 17,297,280 possible combinations for a matchday in a 14-team league, such as the Danish Superliga.
• The rules for the FIFA World Cup demand, that any group contains four teams, of which one or two are UEFA members, and the remaining teams are from different non-UEFA confederations. The number of possible combinations is currently equal to 25,769,535.
• There are 16!/8! = 30!!!! = 518,918,400 possible ways to draw the round of 16 of a single-elimination tournament.
• For the round of 32 of the UEFA Cup, the eight group winners had to be drawn against a third-placed team of another group, and the eight runner-ups had to be drawn against one of the eight third-placed teams from the UEFA Champions League. The number of possible outcomes was !8 × 8! = 598,066,560. In reality, the actual number would be almost always smaller (and was always smaller), since teams of the same association could not be drawn against each other.
• Since there are 211 FIFA members, and at most one of the 11 OFC members may qualify for the FIFA World Cup, there are 200P4 + 4 × 11 × 200P3 = 1,899,176,400 possible combinations for the four highest ranked nations.
• Ignoring chronological order, there are 18!/9! = 34!!!! = 17,643,225,600 possible combinations for a matchday in an 18-team league, such as the German Bundesliga.
• Ignoring chronological order, there are 20!/10! = 38!!!! = 670,442,572,800 possible combinations for a matchday in a 20-team league, such as the English Premier League.
• Ignoring the exact ranking of teams eliminated before the semifinals, there are 614 × 24 = 615 × 4 = 1,880,739,938,304 possible final rankings for the FIFA World Cup after the draw of the group stage.
• For the round of 32 of the UEFA Europa League, the twelve group winners and the four best third-placed teams from the UEFA Champions League have to be drawn against a runner-up of another group or one of the four other third-placed teams from the UEFA Champions League. The number of possible outcomes is !12 + 4 × !13 + 6 × !14 + 4 × !15 + !16 = 9,823,096,307,544. In reality, the actual number is almost always smaller, since teams of the same association cannot be drawn against each other.
• For the round of 32 of the UEFA Women's Champions League, the sixteen seeded teams have to be drawn against the sixteen unseeded teams. The number of possible outcomes is 16! = 20,922,789,888,000. In reality, the actual number is almost always smaller, since teams of the same association cannot be drawn against each other.
• Ignoring chronological order, there are 22!/11! = 42!!!! = 28,158,588,057,600 possible combinations for a matchday in a 22-team league, such as the Spanish Segunda División.
• Ignoring chronological order, there are 24!/12! = 46!!!! = 1,295,295,050,649,600 possible combinations for a matchday in a 24-team league, such as the EFL Championship.
• There are 32!/16! = 62!!!! = 32P16 = 12,576,278,705,767,096,320,000 possible ways to draw the round of 32 of a single-elimination tournament (e.g 16 out of 32 rounds).
• There are 32!/60 = 4,385,513,948,894,892,169,453,633,536,000,000 possible ways to distribute the flags of 32 nations on the 32 faces of a football.

## Approximations of these numbers

### Class 1

42,195:

Notation Approximation
Scientific notation $$4.2195 \times 10^4$$ (exact)
Up-arrow notation $$205↑2 < n < 206↑2$$
BEAF $$\{205,2\} < n < \{206,2\}$$
Chained arrow notation $$205→2 < n < 206→2$$
Fast-growing hierarchy $$f_1(21,097) < n < f_1(21,098)$$
Hardy hierarchy $$H_{\omega}(21,097) < n < H_{\omega}(21,098)$$
Slow-growing hierarchy $$g_{\omega^{205}}(2) < n < g_{\omega^{206}}(2)$$