Math / Numbers & Operations

QuestionFactorial expressions
Evaluate. $\frac{10!6!}{8!8!}$
Simplify your answer a $\square$

Studdy Solution

STEP 1

1. We need to evaluate the factorial expression $\frac{10!6!}{8!8!}$ .
2. Factorials are defined as the product of all positive integers up to a given number.
3. Simplification involves canceling common terms in the numerator and the denominator.

STEP 2

1. Expand the factorials.
2. Simplify the expression by canceling common terms.
3. Calculate the remaining product.

STEP 3

Expand the factorials in the expression. Start with the numerator:
$10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
$6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$

STEP 4

Expand the factorials in the denominator:
$8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$

STEP 5

Substitute the expanded factorials into the expression:
$\frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$

STEP 6

Cancel common terms in the numerator and the denominator:
- Cancel $8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ from both $10!$ and one $8!$ . - Cancel $6 \times 5 \times 4 \times 3 \times 2 \times 1$ from $6!$ and the remaining $8!$ .
This simplifies to:
$\frac{10 \times 9}{8 \times 7}$

STEP 7

Calculate the remaining product:
$\frac{10 \times 9}{8 \times 7} = \frac{90}{56}$
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
$\frac{90}{56} = \frac{45}{28}$
The simplified answer is:
$\boxed{\frac{45}{28}}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms