Math Problem Statement
Simplify \( \frac{{^5C_2 \cdot ^4C_1}}{{^{12}C_3}} \) and express the answer as an integer or a simplified fraction.
Solution
To solve this expression, we’ll break it down by evaluating each combination (denoted as ) term and then performing the necessary multiplication and division.
The expression is:
Step 1: Evaluate Each Combination Term
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Calculate :
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Calculate :
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Calculate :
Step 2: Substitute and Simplify
Now, substitute these values into the original expression:
Step 3: Simplify the Fraction
Final Answer:
Would you like further explanation or have any questions?
- What are combinations and how do they differ from permutations?
- How is the formula for combinations derived?
- What are other real-world applications of combinations?
- How would the answer change if we used instead of ?
- Can the combination formula be used for non-integer values?
Tip: For simplification, always check if the numerator and denominator share any common factors. This can save time when reducing fractions.
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Math Problem Analysis
Mathematical Concepts
Combinatorics
Combinations
Fraction Simplification
Formulas
Combination formula: \( ^nC_r = \frac{n!}{r!(n - r)!} \)
Theorems
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Suitable Grade Level
Grades 10-12