Question

How do you simplify $ \dfrac{{12!}}{{8!4!}} $ ?

Answer 1

Hint: The given expression is in the form of factorial. Factorial can be expressed as the product of all the positive integers less than or equal to the given number. For Example if “n” is the positive integer, the n factorial is denoted by $ n! $ it is the product of all positive integers less than or equal to n. It is expressed as $ n! = n(n - 1)(n - 2).....(2)(1) $ Here we will apply the standard formula of factorial both in the numerator and the denominator and then will simplify for the resultant value.

Complete step-by-step answer:
Take the given expression –
 $ \dfrac{{12!}}{{8!4!}} $
Apply the standard formula in the above expression –
 \[ = \dfrac{{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{\{ 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\} \{ 4 \times 3 \times 2 \times 1\} }}\]
Common factors from the numerator and the denominator cancel each other. So remove $ 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $ from the numerator and the denominator.
 \[ = \dfrac{{12 \times 11 \times 10 \times 9}}{{4 \times 3 \times 2 \times 1}}\]
Find out the factors in the numerator part of the above expression –
 \[ = \dfrac{{4 \times 3 \times 11 \times 2 \times 5 \times 9}}{{4 \times 3 \times 2 \times 1}}\]
Common factors from the numerator and the denominator cancel each other. Therefore, remove $ 4 \times 3 \times 2 $ from the numerator and the denominator.
 \[ = \dfrac{{11 \times 5 \times 9}}{1}\]
Simplify the above expression.
 \[ = 11 \times 5 \times 9\]
Find the product of the above numbers
 $ = 495 $
So, the simplified form of the given expression will be $ 495 $
So, the correct answer is “ $ 495 $ ”.

Note: You should be very good in multiples. As, ultimately your answer depends on the multiplication of the numbers. Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers. For Example: $ 2,{\text{ 3, 5, 7,}}...... $ $ 2 $ is the prime number as it can have only $ 1 $ factor.