Question

Count the possible combinations of \(r\) letters chosen from the given list. \(\mathrm{D}, \mathrm{E}, \mathrm{F}, \mathrm{G}, \mathrm{H} ; r=4\)

Short answer

The number of combinations of 4 letters chosen from the given list is 5.

Step-by-step solution

Identify the combination formula

In combinatorics, the formula for combinations when selecting \(r\) items out of a total of \(n\) items (where order does not matter) is \(C(n,r) = \frac{n!}{r!(n-r)!}\), where '!' denotes the factorial, i.e. the product of all positive integers up to that number.

Apply combination formula to our problem

In this problem, we have 5 letters (\(D, E, F, G, H\)) and we are to select 4. So, \(n=5\) and \(r=4\). Plugging these values into our combination formula, we get \(C(5,4) = \frac{5!}{4!(5-4)!}\)

Calculate the factorial values for n, r and (n-r)

We need to calculate factorials for \(n=5\), \(r=4\), and \((n-r)=1\). So: \(5!=5*4*3*2*1=120\), \(4!=4*3*2*1=24\), and \(1!=1\)

Solve for C(5,4)

Substitute the factorial values back into the combination formula: \(C(5,4) = \frac{120}{24*1}\)

Final calculation

Finally, divide the numbers to get the number of combinations: \(C(5,4) = 5\)

Key concepts

Understanding Factorials

Factorials are a fundamental part of mathematics, especially in the context of counting principles like permutations and combinations. The concept of a factorial is represented with an exclamation mark (!).
For any positive integer "n", the factorial, denoted as "n!", is the product of all integers from 1 to "n". For example, 5 factorial, written as 5!, equals 5 multiplied by 4, then by 3, then by 2, and finally by 1, resulting in 120. Hence, 5! = 5 × 4 × 3 × 2 × 1 = 120.

• Factorials grow fast: The value increases very rapidly as the number increases.
• Zero factorial is special: By definition, 0! is 1.

Understanding the use of factorials is key in solving combination problems, like the one above where calculations simpler using these products of sequential numbers.

What is Combinatorics?

Combinatorics is a branch of mathematics that studies the counting, arrangement, and combination of objects. It helps us answer questions about counting with precision and strategy.
In simpler terms, combinatorics lets you know how many different ways you can select items without worrying about arrangements.

For instance:

• If you have a couple of letters, like D, E, F, G, and H, and you want to know how many different selections of four letters you can make—you use combinatorics!
• Combinatorics is not only about choosing items; it’s about understanding how to choose effectively when faced with limits like order or repetition.

Using combinatorics in the given exercise helps us determine that the number of ways to choose 4 letters from a set of 5 is just 5, showing a practical application of this math field.

Explaining the nCr Formula

The nCr formula, spoken as "n choose r", is a formula used to find the number of combinations you can make when choosing "r" items from a set of "n" total items. In mathematical terms, it is written as \( C(n,r) \) or sometimes \( \binom{n}{r} \).
The formula is:\[C(n,r) = \frac{n!}{r!(n-r)!}\]

Here’s a breakdown of what this formula means:

• "n" represents the total number of items you can choose from.
• "r" represents the number of items to choose.
• The formula uses factorials to ensure that the order does not matter in combinations, differing from permutations where order is crucial.

In the example problem, we calculate \( C(5,4) \), where you select 4 letters from 5. Plugging in the numbers, you simplify the equation to 5, revealing the number of possible combinations.