<span title="Solve............................................?">Solve.......................</span>

Question

(n) -n = 27

(2)

treat it as though the n and the 2 are in one bracket with the n over the 2 but not divided by it

Answer 1

The one bracket with n at the top and 2 at the bottom is read as n choose 2, or the number of 2 combinations from a set with n elements.  This can be written as:

C(n, k) = n! / [k!(n - k)!]

In your case, k = 2 so this makes your equation:

n! / [2!(n - 2)!] - n = 27

(n(n - 1)(n - 2)!) / [2(n - 2)!] - n = 27

(n(n - 1))/2 - n = 27

½(n² - n) - n = 27

n²/2  - n/2 - n = 27

n²/2  - 3n/2 - 27 = 0

½(n² - 3n - 54) = 0

½(n - 9)(n + 6) = 0

Then the solutions are either n = 9 or n = -6.  Since you cannot choose 2 items from a set of -6 elements, the answer is n = 9.

Hope this helps you!