Evaluate the given binomial coefficient. (1002)\binom{100}{2}

Ch. 8 - Sequences, Induction, and Probability

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 7

9장, 문제 7

Evaluate the given binomial coefficient. $open paren 100 over 2 close paren$

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Recognize that the binomial coefficient $ \binom{100}{2} $ represents the number of ways to choose 2 items from 100 without regard to order.

Recall the formula for the binomial coefficient: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \], where $ n = 100 $ and $ r = 2 $.

Substitute the values into the formula: \[ \binom{100}{2} = \frac{100!}{2!(100-2)!} = \frac{100!}{2! \cdot 98!} \].

Simplify the factorial expression by canceling $ 98! $ from numerator and denominator: \[ \frac{100 \times 99 \times 98!}{2! \times 98!} = \frac{100 \times 99}{2!} \].

Calculate the denominator $ 2! = 2 $, so the expression becomes \[ \frac{100 \times 99}{2} \]. This is the simplified form ready for evaluation.

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주요 개념

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Binomial Coefficient

The binomial coefficient, denoted as (n choose k), represents the number of ways to choose k elements from a set of n elements without regard to order. It is calculated using the formula n! / (k! (n-k)!), where '!' denotes factorial.

Factorials

A factorial of a non-negative integer n, written as n!, is the product of all positive integers less than or equal to n. Factorials are essential in computing binomial coefficients and appear frequently in permutations and combinations.

Simplification Techniques for Large Factorials

When evaluating binomial coefficients with large numbers, it is efficient to simplify factorial expressions by canceling common terms before multiplying. For example, (100 choose 2) can be simplified by expanding only the necessary terms to avoid large computations.

Evaluate the given binomial coefficient. (1002)\(\binom{100}{2}\)