Question

7.4
score: 8/17 answered: 8/17
question 9
evaluate.
\\(\dbinom{100}{1}\\) =
question help: video ebook

Explanation:

Step1: Recall the combination formula

The binomial coefficient \(\binom{n}{k}\) is calculated using the formula \(\binom{n}{k}=\frac{n!}{k!(n - k)!}\), where \(n!=n\times(n - 1)\times\cdots\times1\) for \(n\geq1\) and \(0!=1\). Here, \(n = 100\) and \(k = 1\).

Step2: Substitute values into the formula

Substitute \(n = 100\) and \(k = 1\) into the formula: \(\binom{100}{1}=\frac{100!}{1!(100 - 1)!}=\frac{100!}{1!×99!}\).
Since \(n!=n\times(n - 1)!\), then \(100!=100\times99!\). So we can rewrite the expression as \(\frac{100\times99!}{1\times99!}\).

Step3: Simplify the expression

The \(99!\) terms in the numerator and denominator cancel out, leaving us with \(100\div1 = 100\).

Answer:

\(100\)