Evaluate 7C6
Understand the Problem
The question is asking us to evaluate the combination formula, which determines how many ways we can choose 6 items from a set of 7. This is done using the formula for combinations, nCr = n! / (r!(n - r)!)
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Answer
$7$
Answer for screen readers
The answer is $7$.
Steps to Solve
- Identify the values for n and r
In this problem, we have $n = 7$ (the total number of items) and $r = 6$ (the number of items to choose).
- Plug values into the combinations formula
Using the combinations formula $nCr = \frac{n!}{r!(n - r)!}$, we can substitute the values we have:
$$ 7C6 = \frac{7!}{6!(7 - 6)!} $$
- Simplify the equation
Now, let's simplify the equation:
$$ 7C6 = \frac{7!}{6! \cdot 1!} $$
- Calculate factorials
Then, we calculate the factorials:
$$ 7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040 $$
$$ 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 $$
$$ 1! = 1 $$
- Substitute the factorial values back into the equation
Now we substitute the calculated factorials back into our combinations formula:
$$ 7C6 = \frac{5040}{720 \cdot 1} $$
- Perform the division
Now, perform the division calculation:
$$ 7C6 = \frac{5040}{720} = 7 $$
The answer is $7$.
More Information
This result means there are 7 different ways to choose 6 items from a set of 7. In simpler terms, choosing 6 out of 7 is the same as choosing 1 item to leave out, hence the number of ways corresponds to the 7 items available.
Tips
- Forgetting that $1! = 1$, which can lead to incorrect simplification of the equation.
- Confusing the order of $n$ and $r$ in the combinations formula.