Questions: Evaluate. 13! / (5!(13-5)!) The answer is . (Type an integer or a simplified fraction.)

$Evaluate. 13! / (5!(13-5)!) The answer is . (Type an integer or a simplified fraction.)$

Transcript text: Quiz: Ready - Prerequisite Week 3 Quiz Question 6 of 25 this quiz: 25 This question Question list Question 1 Question 2 Question 3 Question 4 Question 5 Check here for instructional material to complete this problem. Evaluate. \[ \frac{13!}{5!(13-5)!} \] The answer is . $\square$ (Type an integer or a simplified fraction.)

Solution

Solution Steps

To evaluate the given expression, recognize that it represents a combination formula, specifically "13 choose 5". The formula for combinations is given by $\frac{n!}{k!(n-k)!}$. Here, $n = 13$ and $k = 5$. Calculate the factorial of these numbers and substitute them into the formula to find the result.

Step 1: Identify the Problem

The problem requires evaluating the expression $\frac{13!}{5!(13-5)!}$, which is a combination formula representing "13 choose 5".

Step 2: Apply the Combination Formula

The combination formula is given by: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Substitute $n = 13$ and $k = 5$ into the formula: \[ \binom{13}{5} = \frac{13!}{5!(13-5)!} = \frac{13!}{5! \times 8!} \]

Step 3: Calculate Factorials

Calculate the factorials:

$13! = 6227020800$
$5! = 120$
$8! = 40320$

Step 4: Substitute and Simplify

Substitute the factorial values into the combination formula: \[ \binom{13}{5} = \frac{6227020800}{120 \times 40320} \] Simplify the expression: \[ \binom{13}{5} = \frac{6227020800}{4838400} = 1287 \]

Final Answer

$\boxed{1287}$

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