Question

7.6
score: 2/19 answered: 2/19
question 3
evaluate the expression.
\\(\dbinom{9}{7}\\) =
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Explanation:

Step1: Recall the combination formula

The binomial coefficient \(\binom{n}{k}\) (read as "n choose 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, \(n = 9\) and \(k = 7\).

Step2: Simplify the factorials

First, note that \(\binom{9}{7}=\binom{9}{9 - 7}=\binom{9}{2}\) (since \(\binom{n}{k}=\binom{n}{n - k}\)). Now calculate \(\binom{9}{2}=\frac{9!}{2!(9 - 2)!}=\frac{9!}{2!7!}\). Since \(9! = 9\times8\times7!\) and \(2! = 2\times1\), we can substitute these into the formula: \(\frac{9\times8\times7!}{2\times1\times7!}\). The \(7!\) terms cancel out, leaving \(\frac{9\times8}{2\times1}\).

Step3: Perform the multiplication and division

Calculate \(9\times8 = 72\) and \(2\times1 = 2\). Then \(\frac{72}{2}=36\).

Answer:

36