Remember that n! gives the product of all the numbers from 1 through to n.
8! = 8xx7xx6xx5xx4xx3xx2xx1
If you were to have (8!)/(5!) this is the same as:
(8xx7xx6xxcancel(5!))/(cancel(5!)) which cancels to 8xx7xx6
Simplifying manually is long and tedious, but here goes .....
(15!)/(9!6!) + (7!)/(10!5!)" "larr split up the bigger numbers.
Let's look at the first fraction... (15!)/(9!6!)
(cancel15^cancel3xxcancel14^7xx13xxcancel(12)^cancel2xx11xxcancel10^5xxcancel(9!))/(cancel(9!)cancel(6)xxcancel5xxcancel4^cancel2xxcancel3xxcancel2xx1)" "larr cancel
Cancel: 9! and 9!, " "6 " into "12 " twice, " "5 into " 15 thrice,
2 " into " 14 and 10 and 4," " 3 and 3
This leaves (7xx13xx11xx5)/1 = 5005
Similarly for he second fraction:
(7!)/(10!5!)
= (7xx6xxcancel(5!))/(10!xxcancel(5!))
=cancel(7xx6)/(10xx9xx8xxcancel(7xx6)xx5xx4xx3xx2xx1)
=1/(86400)
Adding the two answers gives: 5005 + 1/86400
= 5000 1/86400
