Answer :

Explanation

The factorial of an integer is given by the product of all the integers that are equal or smaller than it. For example, the factorial of 4 is:

[tex]4!=1\cdot2\cdot3\cdot4[/tex]

We must find the value of the following expression with factorials:

[tex]\frac{4!\cdot6!}{2!\cdot5!}[/tex]

The first thing that we can do is expand the four factorials in the expression:

[tex]\frac{4!\cdot6!}{2!\cdot5!}=\frac{1\cdot2\cdot3\cdot4\cdot1\cdot2\cdot3\cdot4\cdot5\cdot6}{1\cdot2\cdot1\cdot2\cdot3\cdot4\cdot5}[/tex]

All the numbers on both the numerator and the denominator are multiplying. This means that if a number appears in the numerator and in the denominator it can be simplified. For example 1 appears twice in the numerator and twice in the denominator so it can be simplified:

[tex]\frac{1\cdot2\cdot3\cdot4\cdot1\cdot2\cdot3\cdot4\cdot5\cdot6}{1\cdot2\cdot1\cdot2\cdot3\cdot4\cdot5}=\frac{1\cdot1}{1\cdot1}\cdot\frac{2\cdot3\cdot4\cdot2\cdot3\cdot4\cdot5\cdot6}{2\cdot2\cdot3\cdot4\cdot5}=\frac{2\cdot3\cdot4\cdot2\cdot3\cdot4\cdot5\cdot6}{2\cdot2\cdot3\cdot4\cdot5}[/tex]

2 appears twice in both the numerator and the denominator so it can be simplified:

[tex]\frac{2\cdot3\cdot4\cdot2\cdot3\cdot4\cdot5\cdot6}{2\cdot2\cdot3\cdot4\cdot5}=\frac{3\cdot4\cdot3\cdot4\cdot5\cdot6}{3\cdot4\cdot5}[/tex]

We have a 3, a 4 and a 5 in the denominator so we can simplify them with a 3, a 4 and a 5 of the numerator:

[tex]\frac{3\cdot4\cdot3\cdot4\cdot5\cdot6}{3\cdot4\cdot5}=\frac{3\cdot4\cdot5}{3\cdot4\cdot5}\cdot\frac{3\cdot4\cdot6}{1}=3\cdot4\cdot6=72[/tex]Answer

Then the answer is 72.

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