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

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

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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

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[/quote]

Hi,

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

When this says n-digit numbers , does it mean the possible sequences of 5 digit numbers when the dice is rolled 5 times, or what does it mean? I'm sorry, but I found the language quite confusing!

Statement 1: 6 ^n=7776

n = 5. Since we get a unique value of n, this statement is sufficient

Statement 2:
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.

Hence, answer is D

Total outcome when a fair dice is rolled n times = 6^n

(1) Given 6^n = 7776 = 6^5
—> n = 5

Sufficient

(2) Probability of getting 6 on every roll of(n-3) is 1/6^(n - 3) = 1/36
—> 1/6^ (n-3) = 1/6^2
—> n - 3 = 2
—> n = 5

Sufficient

IMO Option D

Hi, I feel the way the first statement is worded is rather confusing. Can't it mean that 7776 is the number of different ways we can arrange a n digit number for which we don't know the digits. Thanks

I'm also having a lot of trouble with the wording of statement 1.


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

I am having trouble making the logical jump in the phrase

"The number of different possible sequences of n-digit numbers"

to \(6^n\).

The word sequences throws me off the most. Is it just asking for the different possibilities?
Apologies if the question is ill informed.

" The number of different possible sequences of n-digit numbers when a dice is rolled n times"

The n-digit number is formed by combining the outcomes of 'n' rolls
If the dice was rolled 1 time, sequence of the 1-digit numbers => 1,2,3,4,5,6 => 6 possible outcomes
If the dice was rolled 2 times, sequence of the 2-digit numbers => 11,12,13,14,.......64,65,66 => 6*6 =36 possible outcomes
If the dice was rolled 3 times, sequence of the 3-digit numbers => 111,112,113,...........666 => 6*6*6 = 216 possible outcomes

The dice was rolled n times, sequence of the n-digit numbers => 6^n = 7776

Statement 1) We know that if we roll the dice 1 time the number of sequences possible is 6... if we roll the dice two times the number of sequences is 6^2. This means that we just need to determine if 7776 is a result of 6^X to which the answer is yes and therefore we can determine the x value and therefore determine the number of rolls. In this case 6^5 is equal to 7776 and therefore the number of rolls OR the value of n is equal to 5

Statement 2) The odds of getting a 6 on one roll is 1/6... the odds of getting another 6 on the next role is 1/6... the odds of rolling a six twice in a row is 1/6 * 1/6 which equals 1/36. Therefore we can determine that 1/36 odds occurred in just two rolls and because this occurred "3 times fewer" the total number of rolls OR value of n is equal to 5.

The answer is D - Each statement is sufficient on it's own!