Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Evaluate the given expression. $$ C(10,3) $$

Step-by-Step Solution

Answer

The given expression is \(C(10,3)\), which represents the number of possible combinations when selecting 3 items from a set of 10. Using the combination formula, we have \(C(10, 3) = \frac{10!}{3!(10-3)!}\), which simplifies to \(C(10, 3) = \frac{3,628,800}{30,240}\). The final result is 120 possible combinations.

Step 1: Identify n and r

In this case, the given expression is C(10,3), so we have n = 10 and r = 3.

Step 2: Factorial calculations

Calculate the factorials of n, r, and (n-r): - n! = 10! = 3,628,800 - r! = 3! = 6 - (n-r)! = (10-3)! = 7! = 5,040

Step 3: Apply the combination formula

Insert the values of n, r, and their respective factorials into the combination formula: \[C(10, 3) = \frac{10!}{3!(10-3)!}\]

Step 4: Evaluate the expression

Substitute the factorial values found in step 2 and evaluate the expression: \[C(10, 3) = \frac{3,628,800}{6 \times 5,040} = \frac{3,628,800}{30,240}\]

Step 5: Simplify the result

Divide the numbers to obtain the final result: \[C(10, 3) = 120\] The result is 120 possible combinations when selecting 3 items from a set of 10.

Key Concepts

Combination Formula

In combinatorics, combinations are an essential concept used to find the number of possible ways to choose a subset of items from a larger set without regards to the order of selection. The formula for combinations is denoted by \( C(n, r) \), which represents choosing \( r \) items from \( n \) total items. The mathematical formula is written as:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n! \) denotes the factorial of \( n \), \( r! \) the factorial of \( r \), and \( (n-r)! \) the factorial of the difference between \( n \) and \( r \).

• This formula is pivotal when the order does not matter, distinguishing it from permutations where the order is important.
• Understanding how to apply the combination formula provides valuable tools for calculating probabilities and analyzing statistical data.

Factorials

Factorials are fundamental in combinatorics and mathematical calculations. When you see the symbol \( n! \), it means the product of all positive integers from 1 to \( n \). For example:

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

Factorials grow quickly with increasing \( n \). They help in calculating combinations and permutations by accounting for all possible arrangements of a set.
Keep in mind:

• \( 0! \) is defined as 1, which is a special case for factorials.
• Factorials are used in various fields, including probability, algebra, and calculus.

Permutations

Unlike combinations, permutations consider the order of items. In permutations, each different arrangement counts as a distinct possibility. For permutations of \( n \) items taken \( r \) at a time, the formula is:\[ P(n, r) = \frac{n!}{(n-r)!} \]highlighting how it differs from the combination formula. For example:

• Arranging 3 out of 5 books on a shelf is a permutation problem.
• If the order were not important, it would become a combination problem.

Therefore, permutations are ideal when the sequence is crucial, such as in arrangements, rankings, or sequences.