Data Sufficiency

LUANDATO wrote:For positive integers n and m, is m! < 3^n ?

(1) n = m
(2) n > 3

The OA is E.

I don't understand this DS question. Can any expert explain it for me please? Thanks.

(1) n = m

Certainly insufficient.

Case 1: Say n = m = 1, then m! = 1! = 1 and 3^n = 3^1 = 3. The answer is Yes.
Case 2: Say n = m = 6, then m! = 7! = 1.2.3.4.5.6.7 = 720.7 and 3^n = 3^6 = 3.3.3.3.3.3.3 = 729.3. We see that 720.7 > 729.3. The answer is No.

(2) n > 3

We hve seen Case 2: Say n = m = 6, then m! = 7! = 1.2.3.4.5.6.7 = 720.7 and 3^n = 3^6 = 3.3.3.3.3.3.3 = 729.3. The answer is No.

Case 3: Say n = m = 4, then m! = 7! = 1.2.3.4. = 24 and 3^n = 3^4 = 3.3.3.3 = 81. The answer is Yes. No uniqe answer.

(1) and (2) combined:

Case 2 and Case 3 are applicable here too. Thus, there is no unique answer. Insufficient.

The correct answer: E

Hope this helps!

-Jay

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