Evaluate 7 choose 6 (7c6).
Understand the Problem
The question is asking for the evaluation of the binomial coefficient 7 choose 6, which represents the number of ways to choose 6 elements from a set of 7 elements. This is calculated using the formula for binomial coefficients.
Answer
7
Answer for screen readers
The answer is ( 7 ).
Steps to Solve
- Identify the binomial coefficient formula
The binomial coefficient "n choose r" is represented by the formula:
$$ C(n, r) = \frac{n!}{r!(n - r)!} $$
Where $n$ is the total number of items, $r$ is the number of items to choose, and $!$ denotes factorial.
- Plug in the values for 7 choose 6
In this case, we have $n = 7$ and $r = 6$. Substitute these values into the formula:
$$ C(7, 6) = \frac{7!}{6!(7 - 6)!} = \frac{7!}{6! \cdot 1!} $$
- Calculate the factorials
Now, compute the factorials:
- $7! = 7 \times 6!$
- $6! = 720$
- $1! = 1$
So, we can rewrite the equation:
$$ C(7, 6) = \frac{7 \times 6!}{6! \cdot 1} $$
- Simplify the expression
Since $6!$ appears in both the numerator and denominator, we can cancel it out:
$$ C(7, 6) = \frac{7}{1} = 7 $$
The answer is ( 7 ).
More Information
The outcome shows that there are 7 different ways to choose 6 elements from a set of 7 elements. This is a common scenario in combinatorics, especially when you need to find subsets of a larger group.
Tips
- A common mistake is not reducing the factorial terms properly, which can lead to an incorrect calculation.
- Another mistake is misinterpreting the values of $n$ and $r$, leading to using incorrect numbers in the binomial coefficient formula.
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