Question

Count the possible combinations of \(r\) letters chosen from the given list. \(\mathrm{L}, \mathrm{M}, \mathrm{N}, \mathrm{O} ; r=2\)

Short answer

The possible combinations of choosing 2 letters from the list are 6.

Step-by-step solution

Determine the parameters for the combination formula

Given the list ['L', 'M', 'N', 'O'], let's identify 'n' as the total number of letters in this list. n = 4. Furthermore, it's stated that r, the number of letters to choose, equals 2.

Plug into the combination formula

Plug the recognized values of n and r into the combinations formula. So, \(C(4, 2) = 4! / 2!(4-2)!\)

Calculate the combination

First, calculate both 4! and 2!. Remember that the factorial of a number is the product of that number and all the numbers below it down to 1. Thus, 4! = 4*3*2*1 = 24 and 2! = 2*1 = 2. Consequently, \(C(4, 2) = 24 / [2*(4-2)!]\). To finish this calculation, compute the factorial of (4-2). It equals 2, which gives \(C(4, 2) = 24 / [2*2]\).

Final calculation

Doing the final arithmetic, \(C(4, 2) = 24 / [4]\). Hence, \(C(4, 2)\) equals 6.

Key concepts

Factorials

Factorials are an essential part of combinatorial mathematics. They are represented by the symbol "!" after a number, indicating that you multiply that number by all the whole numbers less than it down to one. For example, the factorial of 4, denoted as \(4!\), is calculated as \(4 \times 3 \times 2 \times 1 = 24\). This concept helps simplify the calculations involved in finding combinations, as it breaks down complex multiplications into simpler steps.

When you encounter a factorial, remember:

• The factorial of 0 is defined to be 1 (\(0! = 1\)).
• Factorials grow very rapidly; even a small increase in number results in a big increase.

Understanding factorials allows you to manipulate and simplify complex mathematical operations into something much more manageable.

Combinatorial Mathematics

Combinatorial mathematics is the branch of mathematics dealing with counting, arrangement, and combination of elements within a set. In essence, it helps us determine how to select objects from a collection, which is essential for problems involving permutations and combinations.

For example, in the problem, where you have four letters ['L', 'M', 'N', 'O'] and need to choose 2, combinatorial mathematics helps calculate how many different combinations can be made. It uses formulas that involve factorials, which help us avoid listing and counting all possibilities manually.

• Combinations refer to selecting items where the order doesn't matter.
• Permutations refer to arrangements where the order does matter.

The study of combinatorial mathematics provides the necessary tools to solve real-world problems by converting complex scenarios into simple calculations, using principles like those seen in combinations.

Binomial Coefficients

Binomial coefficients are a key concept used in combinatorial mathematics to identify the number of ways to choose \(r\) items from a set of \(n\) items without regard to the order. This is represented by the formula:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

In our example, when you need to choose 2 letters from 4, it is represented as \(C(4, 2)\). Here:

• \(n = 4\), the total number of items,
• \(r = 2\), the number of items to choose.

Plugging these into the formula gives us \(\frac{4!}{2!(4-2)!}\). This simplifies finding all the different combinations without considering different orders, which is crucial in scenarios where arrangement doesn’t matter.

Understanding binomial coefficients allows you to express complex selection problems in simple mathematical terms. It applies broadly, from simple problems like picking letters from a list to advanced probability and statistics.