SOLVED:Evaluate Each Expression. C(12,4) \cdot C(8,3) - Numerade

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Evaluate each expression. $C(12,4) \cdot C(8,3)$ Evaluate each expression.$C(12,4) \cdot C(8,3)$ Algebra 2 Holliday, Luchin,… 1st Edition Chapter 12, Problem 19 View All Chapters

Step 1

Using the combination formula, we have \[C(12,4) = \frac{12!}{(12-4)!4!} = \frac{12!}{8!4!}\] Expanding the factorials, we get \[C(12,4) = \frac{12 \times 11 \times 10 \times 9 \times 8!}{8! \times 4 \times 3 \times 2 \times 1}\] We can cancel out the $8!$ in Show more…

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Key Concepts

Factorial Function The factorial function, denoted by n!, is a fundamental concept in mathematics, particularly in combinatorics. It represents the product of all positive integers from 1 up to n. Factorials are used in the calculation of combinations and permutations, serving as critical components in the formulas that determine the number of ways items can be arranged or selected. Multiplicative Principle The multiplicative principle, also known as the rule of product in combinatorics, states that if one event can occur in a ways and a second independent event can occur in b ways, then the total number of ways both events can occur is a × b. This principle is useful for combining the outcomes of multiple stages or processes, such as when evaluating the product of two separate combination expressions. Combination Combination refers to the process of selecting items from a larger set where the order of selection does not matter. This concept is central to combinatorial calculations and is typically represented using the binomial coefficient notation, often written as C(n, k) or 'n choose k'. The formula for a combination is given by n! / (k!(n-k)!), which quantifies the number of distinct subsets of k items that can be drawn from a set of n items.

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Transcript

00:04 In this problem, we're multiplying two combinations together, and perhaps you're going to use your calculator to determine each combination and then multiply those answers. 00:13 Or perhaps you're going to use your factorial button on your calculator. 00:16 I'll show what we would do if we're not using either button. 00:20 So using the combination factoria for combination formula for 12 choose 4, we would have 12 factorial over 12 minus 4 factorial times 4, and that works out to be 12 factorial over 8 factorial times 4 factorial. 00:42 And if you expand those, you have 12 times 11 times 10 times 9 times 8 factorial over 8 factorial times 4 times 3 times 2 times 1. 00:55 So i only expanded as much as necessary, but i kept some things as factorials to make it simpler. 01:01 That way i could cancel the 8 factorials on the top and the bottom. 01:04 Also notice that we have a 4 times 3 on the bottom that would cancel with the 12 on the top. 01:10 And now we're left with 11 times 10 times 9 over 2... Need help? Use Ace Ace is your personal tutor. It breaks down any question with clear steps so you can learn. Start Using Ace Ace is your personal tutor for learning Step-by-step explanations Instant summaries Summarize YouTube videos Understand textbook images or PDFs Study tools like quizzes and flashcards Listen to your notes as a podcast

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