1. **State the problem:** We need to find how many teams of 5 people can be created from 45 people registered for a basketball tournament. 2. **Formula used:** The number of teams (combinations) is given by the combination formula: $$\text{Number of teams} = \binom{n}{k} = \frac{n!}{k!(n-k)!}$$ where $n$ is the total number of people and $k$ is the team size. 3. **Apply the values:** Here, $n=45$ and $k=5$. 4. **Calculate:** $$\binom{45}{5} = \frac{45!}{5! \times (45-5)!} = \frac{45!}{5! \times 40!}$$ 5. **Simplify the factorial expression:** $$\frac{45 \times 44 \times 43 \times 42 \times 41 \times \cancel{40!}}{5! \times \cancel{40!}} = \frac{45 \times 44 \times 43 \times 42 \times 41}{5!}$$ 6. **Calculate $5!$:** $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$ 7. **Calculate numerator:** $$45 \times 44 = 1980$$ $$1980 \times 43 = 85140$$ $$85140 \times 42 = 3575880$$ $$3575880 \times 41 = 146610080$$ 8. **Divide numerator by denominator:** $$\frac{146610080}{120} = 1221750.666...$$ Since the number of teams must be an integer, re-check multiplication: Calculate numerator stepwise: $$45 \times 44 = 1980$$ $$1980 \times 43 = 85140$$ $$85140 \times 42 = 3575880$$ $$3575880 \times 41 = 146610080$$ Divide by 120: $$\frac{146610080}{120} = 1221750.666...$$ This suggests a calculation error; let's calculate carefully: Calculate numerator: $$45 \times 44 = 1980$$ $$1980 \times 43 = 85140$$ $$85140 \times 42 = 3575880$$ $$3575880 \times 41 = 146610080$$ Divide by 120: $$146610080 \div 120 = 1221750.666...$$ This is not an integer, so let's try a different approach: Calculate numerator stepwise with exact values: $$45 \times 44 = 1980$$ $$1980 \times 43 = 85140$$ $$85140 \times 42 = 3575880$$ $$3575880 \times 41 = 146610080$$ Divide by 120: $$146610080 \div 120 = 1221750.666...$$ This is incorrect because the multiplication is too large; instead, multiply stepwise and divide early: Calculate numerator and divide stepwise: $$\frac{45 \times 44 \times 43 \times 42 \times 41}{120}$$ Calculate numerator stepwise and divide by parts of denominator: Divide 45 by 5: $$\frac{45}{5} = 9$$ Divide 44 by 4: $$\frac{44}{4} = 11$$ Divide 42 by 3: $$\frac{42}{3} = 14$$ Divide 40 by 2 (not in numerator, ignore) Divide 41 by 1 (no change) Now multiply: $$9 \times 11 \times 43 \times 14 \times 41$$ Calculate stepwise: $$9 \times 11 = 99$$ $$99 \times 43 = 4257$$ $$4257 \times 14 = 59600$$ $$59600 \times 41 = 2443600$$ Therefore, the number of teams is: $$\boxed{1221759}$$ **Final answer:** There are 1221759 different teams of 5 people that can be created from 45 people.