Question

Write the following products in factorial notation: $6\times7\times8\times9\times10\times11\times12$.

Answer 1

Hint:
We know that a factorial of a number n is given by $n\times(n-1)\times(n-2)\times…\times3\times2\times1$.So by multiplying and dividing our given number by 5! we get a product which can be further written in the required factorial form.

Complete step by step solution:
We are given a product $46\times7\times8\times9\times10\times11\times12$
We are asked to write it in a factorial notation
A factorial of n is given by $n\times(n-1)\times(n-2)\times…..\times3\times2\times1$
The given product can be written as $12\times11\times10\times9\times8\times7\times6$
Now let's multiply and divide by 5!
$
   \Rightarrow 12\times11\times10\times9\times8\times7\times6\times\dfrac{{5!}}{{5!}} \\
   \Rightarrow \dfrac{{12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}}{{5!}} \\
$
Hence here our numerator can be written as 12!
$ \Rightarrow \dfrac{{12!}}{{5!}}$

Hence we have written the given number is factorial notation.

Note:
A) Factorials are always integers because it's the result of multiplying integers together.
B) A common mistake that students make is doing something like:
$\dfrac{{4!}}{{2!}} = \dfrac{{2!}}{{1!}} = 2$
Which is very tempting to do, because they look just like a fraction. However, if we expand the terms, we will see that:
$\dfrac{{4!}}{{2!}} = \dfrac{{4\times3\times2\times1}}{{2\times1}} = 12$ which is different