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Magazine Chapter 5: Problem 39
Calculate the given combination.$$_{10} C_{4}$$
Short Answer
Expert verified
The combination \( _{10}C_4 \) equals 210.
Step by step solution
01
Understanding Combinations
Combinations are used to determine how many ways you can choose a subset of items from a larger set. The formula for calculating combinations is given by \( _nC_r = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \( ! \) denotes factorial, which is the product of all positive integers up to that number.
02
Substitute the Values into the Formula
In this exercise, we need to calculate \( _{10}C_4 \). Using the formula, substitute \( n = 10 \) and \( r = 4 \): \[ _{10}C_4 = \frac{10!}{4!(10-4)!} \].
03
Calculate the Factorials
Compute the factorials needed for the formula: - Calculate \( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800 \).- Calculate \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).- Calculate \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \).
04
Plug Factorials into the Formula and Simplify
Substitute the factorial values back into the formula: \[ _{10}C_4 = \frac{3628800}{24 \times 720} \]. Simplify the denominator first: \( 24 \times 720 = 17280 \). Now divide the numerator by the simplified denominator: \( \frac{3628800}{17280} = 210 \).
05
Final Result
The number of ways to choose 4 items from a set of 10 is \( 210 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Factorials
Factorials are a fundamental concept in mathematics, especially when dealing with permutations and combinations. The factorial of a number, denoted by an exclamation mark \(n!\), is the product of all positive integers from 1 to that number.
- For example, 5! = 5 脳 4 脳 3 脳 2 脳 1 = 120.
- This concept helps simplify expressions in combinatorial calculations.
Exploring Combinatorics
Combinatorics is a branch of mathematics focused on counting and arranging objects. It's an exciting field that finds applications in various problems involving configurations, such as organizing people, arranging schedules, or selecting items.
- The primary goal in combinatorics is to determine how many such arrangements or selections are possible given specific constraints.
- It's divided into two main types: permutations, where order matters, and combinations, where order does not matter.
Insight into Probability
Probability is a measure of how likely an event is to occur. In combination with combinatorics, probability helps us understand more complex phenomena like games of chance, weather forecasts, and risk assessments.
- It's expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
- A common expression is that of calculating the probability of an event as the ratio of favorable outcomes to the total possible outcomes, using the formula: \( P(E) = \frac{ \text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
Deciphering Subset Selection
Subset selection is a concept that involves picking a smaller group of items from a larger set. It's often necessary in decision-making processes, selections, and allotments.
- Choosing subsets relates to combinations since you determine how many ways you can select elements without regard to the order.
- The formula for computing combinations, \( _nC_r = \frac{n!}{r!(n-r)!} \), is directly linked to subset selection.
