Problem 24 - Algebra and Trigonometry

Question

Evaluate each expression. \(1-\frac{_{5} P_{3}}{_{10} P_{4}}\)

Step-by-Step Solution

Answer

First calculate the permutations: \(_5P_3 = \frac{120}{2}=60\), \(_{10}P_4 = \frac{3628800}{720}=5040\). Then subtract and divide: \(1-\frac{60}{5040} = 1 - 0.0119 = 0.9881 \)

Step 1 - Evaluate first permutation

Evaluate the first permutation term \(_5P_3 = \frac{5!}{(5-3)!}\). Calculate 5! and 2!. Then divide 5! by 2!.

Step 2 - Evaluate second permutation

Evaluate the second permutation term \(_{10}P_4 = \frac{10!}{(10-4)!}\). Calculate 10! and 6!. Then divide 10! by 6!.

Step 3 - Subtract the terms

Subtract the first permutation from 1 and then divide the result by the second permutation to get the final answer.

Key Concepts

Factorial Notation

Factorial notation plays a critical role in counting and probability exercises. It is expressed as an integer followed by an exclamation mark (e.g., 5!, read as 'five factorial'). The factorial of a non-negative integer is the product of all positive integers less than or equal to that integer. Easier put, if we have a number, say 4, then the factorial of 4, written as 4!, is calculated by multiplying 4 by every number below it (i.e., 4 x 3 x 2 x 1), which equals 24.

Another important aspect to remember is that the factorial of 0, denoted as 0!, is always 1, by definition. Understanding this principle is essential when evaluating expressions that involve factorials, such as those that show up in various permutation problems.

Permutations

When we delve into permutations, we are essentially exploring the different ways to arrange a set of objects. Permutations only concern arrangements where the order matters. In mathematical terms, a permutation is denoted as _ambda{} P\text kappa{}, where _ is the total number of objects to choose from, and \text kappa{} is the number of objects we want to arrange.

For example, if you have a library of 5 books and you want to see how many unique ways you can arrange 3 of them on a shelf, you would use the permutation formula \( _5P_3 = \frac{5!}{(5-3)!} \) to get your answer. This formula incorporates the concept of factoring out the arrangements of the remaining objects that are not being considered (in this case, the remaining 2 books), hence the division by \((5-3)!\).

Evaluating Expressions

Evaluating expressions with permutations requires a systematic approach. To evaluate such expressions, you first calculate the factorial values involved in the permutation equations. For example, in the step-by-step solution, you first find \(5!\) and \(2!\), and then perform a division. Similarly, to evaluate \(_{10}P_4\), you calculate \(10!\) and \(6!\) before dividing them.

Remember to follow the order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). This ensures accuracy when performing calculations involving multiple steps or operations.

Probability

Probability is the measure of the likelihood that an event will occur, and it can be calculated using permutations, especially when concerned with the possible arrangements of objects or outcomes. Probability values range from 0 (impossibility) to 1 (certainty), and they can be expressed as fractions, decimals, or percentages.

When permutations are used to determine probability, one often divides the number of favorable arrangements (permutations) by the total number of possible arrangements. For example, if you're interested in the probability of drawing a certain hand in a card game, you would evaluate the number of ways to get that hand (favorable outcomes) and then put it over the total number of possible card hand combinations (the sample space). Understanding permutations is thus crucial when calculating the probabilities of various ordered events.