8 choose 0
Understand the Problem
The question refers to the mathematical concept of combinations, specifically choosing 0 items from a total of 8 items. This involves calculating the binomial coefficient, which is commonly represented as 'n choose k'. In this case, '8 choose 0' is equal to 1, because there is exactly one way to choose nothing from a set.
Answer
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Answer for screen readers
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Steps to Solve
- Understanding Combinations
We want to calculate the combination of choosing 0 items from a total of 8 items. This is represented by the binomial coefficient $$ C(n, k) $$ or $$ \binom{n}{k} $$, where $n$ is the total number of items and $k$ is the number of items to choose.
- Set the values for the binomial coefficient
Here, $n = 8$ and $k = 0$. The formula for combinations is given by:
$$ \binom{n}{k} = \frac{n!}{k!(n-k)!} $$
- Substitute the values into the formula
Substituting in our values gives us:
$$ \binom{8}{0} = \frac{8!}{0!(8-0)!} $$
- Calculate Factorials
Calculate the factorials in the equation:
- $8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320$
- $0! = 1$ (by definition)
- $(8-0)! = 8! = 40320$
Now we have:
$$ \binom{8}{0} = \frac{40320}{1 \times 40320} = 1 $$
- Conclusion
Thus, the number of ways to choose 0 items from 8 items is 1, confirming that there is one way to do nothing.
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More Information
In combinatorial mathematics, the number of ways to choose zero items from a set is always one, indicating that there's only one way to not choose anything at all! This is useful in various probability and combinatorial problems.
Tips
- Forgetting that $0! = 1$. This is a common misconception. It's important to remember that the factorial of zero is defined as 1.
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