If a and b are positive integers less than 10, what is the mode of the GMAT Data Sufficiency

To solve this problem, let's analyze each statement individually and then evaluate their combined information.

Statement (1):
The number of different permutations of the numbers in the list is 12.
From the given list, {a, b, 1, 2}, we need to determine the mode.
The mode is the value that appears most frequently in the list.
S1 provides information about the number of different permutations, which is not directly related to finding the mode.
The statement alone does not provide any information about the specific values of a and b or their frequencies in the list.
Therefore, S1 alone is not sufficient to answer the question.

Statement (2):
A four-digit number 21ab is divisible by 9.
For a number to be divisible by 9, the sum of its digits must be divisible by 9.
The four-digit number 21ab can be expressed as 2100 + 10a + b.
The sum of its digits is (2 + 1 + 0 + 0) + (a) + (b).
For the entire expression to be divisible by 9, the sum of its digits must be divisible by 9.
However, this statement does not provide any direct information about the mode or the frequencies of the numbers in the list.
It only provides a condition for divisibility by 9.
Therefore, S2 alone is not sufficient to answer the question.
Combining the statements:
By combining the statements, we still don't have enough information to determine the mode.
The first statement provides information about permutations, while the second statement provides a condition for divisibility by 9.
However, neither statement gives us direct information about the mode or the frequencies of the numbers in the list.
Therefore, the statements together are still not sufficient to answer the question.
Correct option: E

Statement (1)
The number of different permutations of the numbers in the list is 12.
All four cannot be different, otherwise permutations would be 4! = 4 * 3 * 2 = 24
Since there are 12 permutations, and 12=24/2=4!/2=4!/2!
So there is one pair the same and the other two distinct..
But mode could be any 1, or 2 or a variable
Insufficient

Statement (2)
A four-digit number 21ab is divisible by 9
Since divisibility depends on sum of the number, so a+b+2+1=3+a+b is div by 9
Since a be B are interchangeable, we cannot find the value of variable
Number could 2115, 2151, 2124, 2142, 2133, 2169, 2196, 2178, 2187
So, insufficient
Combined
One pair and other two distinct, so possibilities 2115, 2124, 2133
So, mode can be 1,2 or 3
Insufficient
Correct option: E

Statement (1):
The number of different permutations of the numbers in the list is 12.
4! = 24 but the given permutation is 12 so 4! / 2! = 12 but 2 numbers can be any between (0 - 10)
Not sufficient

Statement (2):
21ab is divisible by 9 , so number sum 2 + 1 + a + b = 9, for which we have many values
Not sufficient
From (1) & (2) : we won’t get any 1 single value to determine the mode
Correct option: E