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Work Shown:
The exclamation mark indicates we start at the given number and multiply our way down to 1.
16! = 16*15*14*13*12*11*...*3*2*1
14! = 14*13*12*11*...*3*2*1
2! = 2*1
Note how 14*13*12*11*...*3*2*1 is buried in the sequence for 16! and it is entirely 14! expression.
This means we can say
16! = 16*15*14*13*12*11*...*3*2*1
16! = 16*15*14!
Bold terms to show the replacement being done.
This will allow us to cancel
[tex]\frac{16!}{2!*14!} = \frac{16*15*14!}{2!*14!} = \frac{16*15}{2!} = \frac{16*15}{2*1}[/tex]
The 14! expressions divide and cancel out.
Then we simplify completely
[tex]\frac{16*15}{2*1} = \frac{240}{2} = 120[/tex]
Therefore,
[tex]\frac{16!}{2!*14!} = 120[/tex]
Side note: This expression is the result of computing [tex]_nC_r = \frac{n!}{r!(n-r)!}[/tex] for n = 16 and r = 2. This is the combination formula. This is equivalent to n = 16 and r = 14 due to symmetry.
