How do you evaluate 7C2?

To evaluate \(7C2\), we are looking at a combinatorial expression that represents the number of ways to choose 2 items from a set of 7 distinct items. This is often referred to as "7 choose 2." The formula for combinations is given by:

Combination Formula

The general formula for combinations is:

\( nCr = \frac{n!}{r!(n-r)!} \)

In this formula:

• n is the total number of items (in this case, 7).
• r is the number of items to choose (in this case, 2).
• ! denotes factorial, which is the product of all positive integers up to that number.

Applying the Formula

Now, let's plug in the values for \(n\) and \(r\) into the formula:

\(7C2 = \frac{7!}{2!(7-2)!}\)

This simplifies to:

\(7C2 = \frac{7!}{2! \cdot 5!}\)

Calculating Factorials

Next, we need to calculate the factorials:

• \(7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 5040\)
• \(2! = 2 \times 1 = 2\)
• \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\)

Substituting Back

Now, substituting these values back into our expression gives:

\(7C2 = \frac{5040}{2 \times 120}\)

Calculating the denominator:

\(2 \times 120 = 240\)

So now we have:

\(7C2 = \frac{5040}{240}\)

Final Calculation

Dividing \(5040\) by \(240\) yields:

\(7C2 = 21\)

Understanding the Result

This means there are 21 different ways to choose 2 items from a set of 7. This concept is widely applicable in various fields, such as probability, statistics, and even in everyday decision-making scenarios where choices are involved.

In summary, evaluating \(7C2\) involves applying the combination formula, calculating the necessary factorials, and performing the arithmetic to arrive at the answer of 21. This process not only helps in solving combinatorial problems but also enhances your understanding of how to approach similar problems in the future.