7 choose 0
Understand the Problem
The question is asking for the value of the binomial coefficient, specifically for 7 choose 0, which is a mathematical expression indicating the number of ways to choose 0 elements from a set of 7 elements. Mathematically, it is represented as C(7, 0) and it equals 1.
Answer
1
Answer for screen readers
The value of the binomial coefficient ( C(7, 0) ) is 1.
Steps to Solve
- Understanding Binomial Coefficient Formula
The binomial coefficient is defined by the formula: $$ C(n, k) = \frac{n!}{k!(n-k)!} $$ where $n$ is the total number of items, $k$ is the number of items to choose, and $!$ denotes factorial.
- Plugging in the Values
For our problem, we have $n = 7$ and $k = 0$. So we can plug these values into the formula: $$ C(7, 0) = \frac{7!}{0!(7-0)!} $$
- Calculating Factorials
Now we calculate the factorials:
- $7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040$
- $0! = 1$ (by definition)
- $(7-0)! = 7! = 5040$
So substituting these values back into our equation: $$ C(7, 0) = \frac{5040}{1 \times 5040} $$
- Simplifying the Equation
Now simplify the fraction: $$ C(7, 0) = \frac{5040}{5040} = 1 $$
The value of the binomial coefficient ( C(7, 0) ) is 1.
More Information
The binomial coefficient ( C(n, 0) ) is always equal to 1, regardless of the value of ( n ), because there is exactly one way to choose none of the items from a set.
Tips
- A common mistake is to confuse the meaning of ( C(n, 0) ) and think it is zero. Remember that choosing nothing from any set has exactly one way to do it.
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