Problem 20 - Algebra and Trigonometry

Question

Evaluate the expression. $$ C(10,5) $$

Step-by-Step Solution

Answer

The value of \( C(10,5) \) is 252.

Step 1: Understand Combination Formula

To evaluate \( C(10,5) \), we need to use the combination formula. The combination formula is given by \( C(n, r) = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, and \( r \) is the number of items chosen from the total.

Step 2: Substitute Values into the Formula

In this exercise, \( n = 10 \) and \( r = 5 \). Substitute these values into the combination formula: \( C(10,5) = \frac{10!}{5!(10-5)!} \). Simplifying gives \( C(10,5) = \frac{10!}{5!5!} \).

Step 3: Calculate Factorials in the Denominator

Calculate the factorial of numbers in the denominator: \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Step 4: Calculate the Numerator Factorial

Calculate the factorial of 10: \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800 \).

Step 5: Simplify the Fraction

Now substitute the factorial calculations into the combination formula: \( C(10,5) = \frac{3,628,800}{120 \times 120} \). Calculate \( 120^2 = 14,400 \). Thus, \( C(10,5) = \frac{3,628,800}{14,400} \).

Step 6: Divide to Find the Result

Finally, calculate the division: \( 3,628,800 \div 14,400 = 252 \). Thus, \( C(10,5) = 252 \).

Key Concepts

Combination Formula

The combination formula is a fundamental concept in combinatorics. It helps us determine how many ways we can choose a subset of items from a larger set without considering the order of selection. This is different from permutations, where order matters. The combination formula is written as \( C(n, r) \), where \( n \) represents the total number of items, and \( r \) represents the number of items to be chosen.

The formula itself is expressed as:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

This formula allows us to calculate the total number of combinations possible by dividing the factorial of the total items by the product of the factorial of chosen items and the factorial of the difference between total and chosen items.

This is very useful in numerous applications like creating teams from a group of players or picking lottery numbers. When using the combination formula, keep in mind that the order of selection does not change the outcome; choosing item A then B is the same as B then A.

Factorials

Factorials are a core component of combinatorics and the combination formula. A factorial, denoted by \( n! \), is the product of all positive integers up to a certain number \( n \). It's a way of counting how many ways you can arrange a set of \( n \) objects.

Let's look at few examples:

• \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
• \( 3! = 3 \times 2 \times 1 = 6 \)
• \( 0! = 1 \)

Notice that \( 0! \) is defined as 1. This is a special case that makes calculations in combinatorics valid and consistent.

In combination problems, factorials help calculate the total number of arrangements of the chosen items (the denominator) and the arrangements of the entire set of items (the numerator). By dividing these, we find the number of non-ordered selections possible.

Binomial Coefficient

The binomial coefficient is another term for the combination \( C(n, r) \), and it is often encountered in binomial theorem expansions. It tells us how many ways we can choose \( r \) items from \( n \) items without regard to order.

A binomial coefficient is typically represented in notation as \( \binom{n}{r} \), which directly corresponds to the combination formula \( C(n, r) \). This notation is very common in algebra and calculus, especially in expansions like:

• \( (x + y)^n \)

In these expansions, each term can be expressed with a binomial coefficient as part of the expression.

Using the binomial coefficient, we combine the concepts of factorials and combinations to generate coefficients that appear in binomial expansions. These coefficients make it possible to determine the number of ways to select items without considering the order, which is handy in probability and statistics too.