Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo

Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

Problema 62

Capitolo 9, Problema 62

Evaluate the given binomial coefficient 11 8

Guida verificata passo dopo passo

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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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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Special Products - Cube Formulas

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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Factorials

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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Fundamental Counting Principle

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