Question

Find \(_{9} \mathrm{P}_{4}\)

Short answer

Thus, the short answer for \(_{9} \mathrm{P}_{4}\) is 3024.

Step-by-step solution

Identify the values of n and r

For this problem, we have 9 objects and we need to find the possible arrangements for 4 objects, so n=9 and r=4.

Write the formula for permutation

The formula for permutation is given by: \(_{n} \mathrm{P}_{r}=\frac{n!}{(n-r)!}\).

Substitute the values into the formula

Now, substitute n=9 and r=4 into the formula: \(_{9} \mathrm{P}_{4}=\frac{9!}{(9-4)!}\).

Simplify the expression

Continue by simplifying the expression: \(_{9} \mathrm{P}_{4}=\frac{9!}{5!}\).

Calculate the factorials and the result

Calculate the factorials and divide them to get the result: \(_{9} \mathrm{P}_{4} = \frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{5 \times 4 \times 3 \times 2 \times 1} = \frac{362880}{120} = 3024\). So, \(_{9} \mathrm{P}_{4}\) equals 3024, which means there are 3024 different ways to arrange 4 objects from a set of 9 distinct objects.

Key concepts

Factorials

Factorials are a fundamental concept in permutations and combinations. They are used to determine the number of ways to arrange a set of items. Factoriales are denoted by an exclamation mark (!). For example, the factorial of 5 is written as 5! and calculated as:\[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]The concept is simple: multiply all positive integers from 1 up to the given number. Factorials grow very quickly, meaning that even for small numbers, the factorial can be quite large. Notably, 0! is defined to be 1 by convention, which is critical for its use in mathematical formulas.

Permutation Formula

The permutation formula helps us calculate how many different ways we can arrange a subset of items from a larger pool. It is particularly important when the order of selection matters. The formula is represented as:\[ _{n}P_{r} = \frac{n!}{(n-r)!} \]Where:

• \( n \) is the total number of objects.
• \( r \) is the number of objects we are selecting and arranging.
This means you take the total factorial \( n! \) and divide it by the factorial of the difference \((n-r)!\).The subtraction \( n-r \) accounts for the leftover items not being arranged.

Arrangements

Arrangements, in a mathematical context, refer to the different orders in which a set of objects can be organized. This is crucial for understanding permutations, where the arrangement or order matters. For example, arranging 3 objects A, B, and C gives us:

• ABC
• ACB
• BAC
• BCA
• CAB
• CBA

Each different sequence represents a unique permutation, or arrangement. The more objects involved and the more objects you're choosing to arrange, the larger the number of possible permutations.

Combinatorial Mathematics

Combinatorial mathematics is a broad field that studies counting, arrangement, and combination of elements within a set according to specific rules. It is the foundation for permutations and combinations, dealing with finite or discrete quantities. In combinatorics, we solve problems like determining the number of ways to arrange objects or select them from groups. This area underpins much of modern mathematics and its application can be found in computer science, statistics, and even optimization problems. Its real-world use can range from simple everyday querying to complex algorithm designs for efficient problem-solving.