In Exercises 7-14, perform the indicated calculation.
9.8C3

Verified step by step guidance

1. Step 1: Understand the problem. The notation 9.8C3 represents a combination calculation, where we are selecting 3 items from a total of 9.8 items. The formula for combinations is given by: $\frac{n!}{r!\bullet (n-r)!}$, where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
2. Step 2: Substitute the values into the formula. Here, n = 9.8 and r = 3. The formula becomes: $\frac{\left(9.8\right)!}{\left(3\right)!\bullet (9.8-3)!}$.
3. Step 3: Simplify the denominator. Calculate (n - r), which is 9.8 - 3 = 6.8. The denominator now becomes: $\left(3\right)!\bullet \left(6.8\right)!$.
4. Step 4: Factorials for non-integer values like 9.8! and 6.8! are calculated using the Gamma function, which extends the factorial function to real numbers. You would use a calculator or software capable of handling Gamma functions to compute these values.
5. Step 5: Divide the result of 9.8! by the product of 3! and 6.8! to find the combination value. Ensure all calculations are performed accurately using appropriate tools.

Key Concepts

Combinations

Combinations refer to the selection of items from a larger set where the order of selection does not matter. The notation 'nCr' or 'C(n, r)' represents the number of ways to choose 'r' items from 'n' items. This concept is crucial in probability and statistics, especially when determining possible outcomes in scenarios where arrangement is irrelevant.

Factorial

A factorial, denoted as 'n!', is the product of all positive integers up to 'n'. It is used in combinations and permutations to calculate the total arrangements of a set. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is essential for performing calculations involving combinations and permutations.

Binomial Coefficient

The binomial coefficient, often represented as 'C(n, r)' or 'nCr', quantifies the number of ways to choose 'r' elements from a set of 'n' elements without regard to the order of selection. It is calculated using the formula C(n, r) = n! / (r!(n-r)!), which incorporates factorials. This concept is fundamental in combinatorial mathematics and probability theory.