FRESH GMAT CLUB QUESTION:

A fair six-sided dice was rolled n times. What is the value of n?

(1) The number of different possible sequences of n-digit numbers when a dice is rolled n times is 7776.

(2) If the dice has been rolled 3 times fewer, the probability of getting a 6 on every roll would have been 1/36.
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Question : Number of dice = ?

Statement 1: The number of different possible sequences of n-digit numbers when a dice is rolled n times is 7776.

i.e. 6^n = 7776
but 6^5 = 7776 hence
n = 5
SUFFICIENT

Statement 2: If the dice has been rolled 3 times fewer, the probability of getting a 6 on every roll would have been 1/36.[/quote]

Probability of getting a 6 on every role = 1/6
i.e. (1/6)*(1/6) = 1/36
i.e. when 3 fewer roles are thrown then 2 roles are thrown
i.e. Total Number of times the dice is rolled = 2+3 = 5
SUFFICIENT

Answer: option D
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What made me really think about this question is the wording of statement 2. It took me a seconds to understand what it meant.

Let's break this down

Statement 1: This statement is essentially asking for the number of different possibilities that can come when a dice is rolled n times.
The number of possibilities is 7776

Now, if a die is rolled once, the number of possibilities is 6. And if rolled twice, possibilities are 6*6 (i.e. \(6^2\))

Thus, in our case \(6^n\)=7776

Now, its a matter of using basic arithmetic to get value of n which turns out to be 5. Since we get a unique value of n, this statement is sufficient

Statement 2: Now, if we were to roll a dice once, the probability of getting number 6 on it would be 1/6 (number of favourable possible outcomes/total possible outcomes).

Thus, for probability to be 1/36, the die has to be rolled twice with each time giving us number 6 on the face of the die (1/6*1/6)

Now, this number (i.e. 2) is 3 less than n. Thus n=5. Since we get a unique value of n, this statement is sufficient too

Hence, answer is D
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