Note that :
n! = n(n-1)(n-2)(n-3)...(2)(1)
Example :
3! = 3(2)(1) = 6
4! = 4(3)(2)(1) = 24
5! = 5(4)(3)(2)(1) = 120
From a. :
[tex]\begin{gathered} \frac{9!}{3}=\frac{9\times8\times7\times6\times5\times4\times\cancel{3}\times2\times1}{\cancel{3}} \\ =9\times8\times7\times6\times5\times4\times2 \end{gathered}[/tex]which is obviously not equal to 3! = 3 x 2 x 1 = 6
So, "a" is false
From b :
[tex]\begin{gathered} \frac{9!}{8!}=\frac{9\times\cancel{8\times7\times6\times5\times4\times3\times2\times1}}{\cancel{8\times7\times6\times5\times4\times3\times2\times1}} \\ =9 \end{gathered}[/tex]which is also obvious that it is not equal to 9! = 9 x 8 x 7 x ... x 1
So, "b" is also false
From c :
[tex]\begin{gathered} \frac{9!}{4!5!}=\frac{9\times8\times7\times6\times\cancel{5\times4\times3\times2\times1}}{4\times3\times2\times1\times\cancel{5\times4\times3\times2\times1}} \\ =\frac{9\times8\times7\times\cancel{6}}{4\times\cancel{3\times2}\times1} \\ =\frac{9\times8\times7}{4\times1} \\ =\frac{504}{4} \\ =126 \end{gathered}[/tex]Since the result is equal to 126, therefore c. is TRUE
The cancel symbol, it is used when the numerator and the denominator has the same value.
For example :
[tex]\frac{ab}{b}=\frac{a\cancel{b}}{\cancel{b}}=a[/tex]ab/b will result to a
Lets try an example :
when a = 2
b = 3
[tex]\frac{2(3)}{3}=\frac{6}{3}=2[/tex]It is the same as :
[tex]\frac{2\cancel{(3)}}{\cancel{3}}=2[/tex]It is like multiplying 2 x (3/3) = 2 x (1) = 2