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For any positive integer n, n! denotes t...

For any positive integer n, n! denotes the product of all integers from 1 through n, what is the value of `3! (7 - 2)!` ?

A

`2!`

B

`3 !`

C

`5!`

D

`6!`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem \( 3! \times (7 - 2)! \), we will follow these steps: ### Step 1: Calculate \( 3! \) The factorial of a number \( n \), denoted as \( n! \), is the product of all positive integers from 1 to \( n \). For \( 3! \): \[ 3! = 3 \times 2 \times 1 = 6 \] ### Step 2: Calculate \( (7 - 2)! \) First, simplify \( 7 - 2 \): \[ 7 - 2 = 5 \] Now calculate \( 5! \): \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] ### Step 3: Multiply the results from Step 1 and Step 2 Now we need to multiply \( 3! \) and \( (7 - 2)! \): \[ 3! \times (7 - 2)! = 6 \times 120 \] Calculating the product: \[ 6 \times 120 = 720 \] ### Final Answer Thus, the value of \( 3! \times (7 - 2)! \) is: \[ \boxed{720} \]
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