Question: Evaluate the expression. If the answer is not an integer, round to four decimal places. 31! 28!…

Evaluate the expression. If the answer is not an integer, round to four decimal places. 31! 28!

Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The value is not an integer, 31! = . 28! (Round to four decimal places as needed.) B. The value is an integer. 31! = . 28! (Type an integer.)

Solution

To evaluate the expression \(\frac{31!}{28!}\), start by simplifying the factorials. Notice that: \[ 31! = 31 \times 30 \times 29 \times 28! \] Thus: \[ \frac{31!}{28!} = \frac{31 \times 30 \times 29 \times 28!}{28!} \] The \(28!\) in the numerator and the denominator cancel out, leaving: \[ 31 \times 30 \times 29 \] Now calculate the product: \[ 31 \times 30 = 930 \] \[ 930 \times 29 = 26970 \] The result is 26970. Since the value is an integer, the correct choice is B: \[ \frac{31!}{28!} = 26970 \]