Textbook Question
Write the first three terms in the binomial expansion, expressing the result in simplified form. (x-3)^9
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10. Combinatorics & Probability
Combinatorics
2:29 minutesProblem 62
Textbook Question
Evaluate the given binomial coefficient118
Verified step by step guidance1
Step 1: Recall the formula for a binomial coefficient, which is given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items, \( k \) is the number of items chosen, and \( ! \) represents factorial.
Step 2: Identify \( n \) and \( k \) from the problem. Here, \( n = 11 \) and \( k = 8 \). Substitute these values into the formula: \( \binom{11}{8} = \frac{11!}{8!(11-8)!} \).
Step 3: Simplify the denominator \( (11-8)! \). Since \( 11-8 = 3 \), the denominator becomes \( 8! \cdot 3! \). The formula now looks like \( \binom{11}{8} = \frac{11!}{8! \cdot 3!} \).
Step 4: Expand \( 11! \) in terms of \( 8! \) to simplify the calculation. Recall that \( 11! = 11 \cdot 10 \cdot 9 \cdot 8! \). Substitute this into the formula: \( \binom{11}{8} = \frac{11 \cdot 10 \cdot 9 \cdot 8!}{8! \cdot 3!} \).
Step 5: Cancel \( 8! \) from the numerator and denominator, leaving \( \binom{11}{8} = \frac{11 \cdot 10 \cdot 9}{3!} \). Finally, recall that \( 3! = 3 \cdot 2 \cdot 1 \), so the denominator becomes 6. The result can now be calculated by dividing \( 11 \cdot 10 \cdot 9 \) by 6.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Coefficient
A binomial coefficient, denoted as C(n, k) or 'n choose k', represents the number of ways to choose k elements from a set of n elements without regard to the order of selection. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial, the product of all positive integers up to that number.
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Factorial
The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics, particularly in calculating binomial coefficients, as they provide the necessary counts of arrangements.
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Combinatorial Interpretation
The combinatorial interpretation of binomial coefficients provides a way to understand their significance in counting problems. For instance, C(11, 8) can be interpreted as the number of ways to select 8 items from a total of 11, which is equivalent to selecting 3 items to leave out, illustrating the symmetry C(n, k) = C(n, n-k).
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