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)!).

Answer

$7$

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.

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