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If (n+2)!/n!=90, then n=?

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by Max@Math Revolution » Wed Aug 01, 2018 11:28 pm

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[Math Revolution GMAT math practice question]

If (n+2)!/n!=90, then n=?

A. 8
B. 9
C. 10
D. 11
E. 12
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by Brent@GMATPrepNow » Thu Aug 02, 2018 6:11 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If (n+2)!/n! = 90, then n = ?

A. 8
B. 9
C. 10
D. 11
E. 12
(n+2)! = (n+2)(n+1)(n)(n-1)(n-2)....(3)(2)(1)
n! = (n)(n-1)(n-2)....(3)(2)(1)

Given: (n+2)!/n! = 90
Rewrite as: (n+2)(n+1)(n)(n-1)(n-2)....(3)(2)(1)/(n)(n-1)(n-2)....(3)(2)(1) = 90
Simplify: (n+2)(n+1) = 90
At this point, we can simply test the answer choices.

A) n = 8
Take (n+2)(n+1) = 90 and replace n with 8 to get: (8+2)(8+1) = 90
It works!!
Answer: A

Alternatively, we can start with (n+2)(n+1) = 90
Expand to get: n² + 3n + 2 = 90
Subtract 90 from both sides to get: n² + 3n - 88 = 0
Factor: (n + 11)(n - 8) = 0
So, EITHER n = -11 OR n = 8
Since we can't have negative factorials, it must be the case that n = 8
Answer: A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Sun Aug 05, 2018 5:54 pm
=>

(n+2)!/n!= (n+2)(n+1) = 90 = 10*9.
Thus, n + 2 = 10 or n = 8.

Therefore, the answer is A.
Answer : A
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