Question

Convert into factorial: $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15$.

Answer 1

Hint: We first discuss the concept of factorial. We try to form the given multiplication starting from 1. We multiply the remaining numbers to and also divide them to keep the main expression intact. We convert the multiplication form starting from 1 into their respective factorial form.

Complete step by step answer:
The given multiplication is to be converted to the factorial form. The use for the factorial function is to count how many ways you can choose things from a collection of things.
We know the term $n!$ defines the notion of multiplication of first n natural numbers.
This means $n!=1\times 2\times 3\times ....\times n$.
But the given multiplication $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15$ does not start from 1.
Therefore, we multiply the terms from 1 to 6 to $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15$.
We also divide them to balance the number.
So, $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15=\dfrac{\left( 1\times 2\times ...\times 6 \right)\times \left( 7\times 8\times ......\times 15 \right)}{1\times 2\times ...\times 6}$.
We can see that the numerator is the multiplication of the first 15 natural numbers and the denominator is the multiplication of the first 6 natural numbers.
Therefore, $1\times 2\times ......\times 15=15!$ and $1\times 2\times ......\times 6=6!$.
We get $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15=\dfrac{15!}{6!}$.
Converting $7\times 8\times 9\times 10\times 11\times 12\times 13\times 14\times 15$ into factorial form we get $\dfrac{15!}{6!}$.

Note: these factorials are mainly used in cases of permutation or combination. In case of combination the simplified form of the mathematical expression ${}^{n}{{C}_{r}}$ is ${}^{n}{{C}_{r}}=\dfrac{n!}{r!\times \left( n-r \right)!}$. In case of permutation the simplified form of the mathematical expression ${}^{n}{{P}_{r}}$ is ${}^{n}{{P}_{r}}=\dfrac{n!}{\left( n-r \right)!}$. They are also used in probabilities.