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## List of numbers appearing in sports-related combinatorics Edit

• 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.
• 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.
• The marathon goes over 42,195 metres or 42.195 km or 26.22 miles. It is equal to 291C2.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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).

## Approximations of these numbersEdit

### Class 1 Edit

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 $$g_{\omega^{205}}(2) < n < g_{\omega^{206}}(2)$$