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Statement (1) does not give any useful information that i can see

(n+1)!-(n-1)! is greater than (n+1)!/(n-1)! for any positive number except 1 (as (n-1)! would be 0...i think!...thus making (n+1)!/(n-1)! undefined).
If n is divisible by 2...the lowest possible value of n is two...
this is sufficient to say (n+1)!-(n-1)! > (n+1)!/(n-1)!

Statement (1) does not give any useful information that i can see

(n+1)!-(n-1)! is greater than (n+1)!/(n-1)! for any positive number except 1 (as (n-1)! would be 0...i think!...thus making (n+1)!/(n-1)! undefined). If n is divisible by 2...the lowest possible value of n is two... this is sufficient to say (n+1)!-(n-1)! > (n+1)!/(n-1)!

thus, my answer is B

the whole equation reduces to (n-1)! * (n(n+1) -1) ... this needs to be compared with (n+1)!/(n-1)! = n(n+1)

I) Square root of n is a positive integer
n can be 1 or 2 or any number greater than 2

for n =1 0! =1 (not zero) - LHS will be less than RHS
for n = 2, LHS < RHS
for n - 3, LHS = 2*11 ; RHS = 12 => LHS > RHS
and for any number greater than 2, LHS > RHS

We need (2) to determine that n is greater than 2

(2) only this is not sufficient either as n can be equal to 2

I will go with C too
The equation reduces to (n-1)![n(n+1) -1] > n(n+1)
From statement 1: n can be 1, 4, 9,...
when n is 1: (n-1)![n(n+1) -1] is not > than n(n+1)
when n is 4: (n-1)![n(n+1) -1] is > than n(n+1)
when n is 9: (n-1)![n(n+1) -1] is > than n(n+1)
so statement 1 is not sufficient

From statement 2: n can be any even number such as 2, 4, 6, 8....
when n is 2: (n-1)![n(n+1) -1] is not > than n(n+1)
when n is 4: (n-1)![n(n+1) -1] is > than n(n+1)
so statement 2 is not sufficient

But both put together: n can be 4, 16, .....
As shown above the condition is satisfied for any number greater than 2
Hence C

Statement 1, it repets nothing but the original condition that n is a positive integer. It behaves differently for 1 and any higher positive integer. Hence Insufficient. Hence rule out option D as well.

Statement 2, says n is even. substraction operation on two positive integer ( Large positive integer- small positive integer) is always greater than division operation on of the same two numbers ( large positive number/small positive number). Hence B is sufficient.

substraction operation on two positive integer ( Large positive integer- small positive integer) is always greater than division operation on of the same two numbers ( large positive number/small positive number).

3 - 2 is not greater than 3 /2

Clarification: I meant in question's context (with factorial ). we will never have a situation 3-2 and 3/2 according to statement 2.
Please let me know why not B is correct as per my logic.

When n<=2, (n-1)!<=1
(n-1)!*[n(n+1)-1]-n(n+1)<=n(n+1)-1-n(n+1)=-1<0

When n>=3, (n-1)!>=2
(n-1)!*[n(n+1)-1]-n(n+1)>=2n(n+1)-1-n(n+1)=n(n+1)-1>0

(1) n is square of an integer, n could be 1 or 4 or greater, insufficient
(2) n is even, n could be 2 or 4 or greater, insufficient
Combined, n could only be 4 or greater, sufficient. (It's always >0 when n>=3)

(C) _________________

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