4. Probability
Counting
2:07 minutesProblem 3.RE.44
Textbook Question
In Exercises 41-44, perform the indicated calculation.
44. (5C3)/(10C3)
Verified step by step guidance1
Step 1: Understand the problem. The problem involves calculating a ratio of combinations. Specifically, you need to compute \( \frac{{5C3}}{{10C3}} \), where \( nCk \) represents the number of ways to choose \( k \) items from \( n \) items without regard to order.
Step 2: Recall the formula for combinations. The formula for \( nCk \) is \( \binom{n}{k} = \frac{{n!}}{{k!(n-k)!}} \), where \( n! \) is the factorial of \( n \).
Step 3: Compute \( 5C3 \) using the formula. Substitute \( n = 5 \) and \( k = 3 \) into the formula: \( \binom{5}{3} = \frac{{5!}}{{3!(5-3)!}} = \frac{{5!}}{{3! \cdot 2!}} \). Simplify this expression.
Step 4: Compute \( 10C3 \) using the formula. Substitute \( n = 10 \) and \( k = 3 \) into the formula: \( \binom{10}{3} = \frac{{10!}}{{3!(10-3)!}} = \frac{{10!}}{{3! \cdot 7!}} \). Simplify this expression.
Step 5: Divide the results of \( 5C3 \) and \( 10C3 \). Use the simplified values from Steps 3 and 4 to compute \( \frac{{5C3}}{{10C3}} \). Simplify the fraction to get the final result.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Combinations
Combinations refer to the selection of items from a larger set where the order does not matter. The notation 'nCr' represents the number of ways to choose 'r' items from 'n' items, calculated using the formula n! / (r!(n-r)!), where '!' denotes factorial. Understanding combinations is essential for solving problems involving group selections.
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Combinations
Factorial
A factorial, denoted as 'n!', is the product of all positive integers up to 'n'. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorial calculations, as they help determine the total arrangements or selections of items in various scenarios.
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Probability
Probability is a measure of the likelihood of an event occurring, expressed as a ratio of favorable outcomes to the total number of possible outcomes. In the context of combinations, the ratio of two combinations, such as (5C3)/(10C3), can represent the probability of selecting a specific group from a larger set, providing insights into comparative likelihoods.
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