Question Simplify $7!$ and express the result as an integer or fraction.

Studdy Solution

STEP 1

Assumptions
1. We need to simplify the factorial expression $7!$ .
2. A factorial of a non-negative integer $n$ , denoted by $n!$ , is the product of all positive integers less than or equal to $n$ .

STEP 2

Definition of factorial for a positive integer $n$ is:
$n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1$

STEP 3

Apply the factorial definition to $7!$ :
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

STEP 4

Calculate the product step by step:
$7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

STEP 5

First, multiply the largest two numbers:
$7 \times 6 = 42$

STEP 6

Continue multiplying the result by the next largest number:
$42 \times 5 = 210$

STEP 7

Continue the multiplication process:
$210 \times 4 = 840$

STEP 8

Multiply the result by the next number:
$840 \times 3 = 2520$

STEP 9

Finally, multiply by the last two numbers:
$2520 \times 2 = 5040$

STEP 10

Since the last number to multiply is 1, the result remains the same:
$5040 \times 1 = 5040$

STEP 11

The simplified result of $7!$ is:
$7! = 5040$
Therefore, $7!$ simplifies to the integer 5040.

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