N is a four digit number. N = ABCD where A, B, C, D are distinct

Question

N is a four-digit number. N = ABCD where A, B, C, D are distinct single-digit whole numbers. How many possible values of N exist for any fixed set of four values of A, B, C, D?

(1) A + B + C + D = 10

(2) A + B + C - D = 8

Pietrus95 · Answer

Can someone help with this problem??

I am thinking that both cases can lead to understanding how many sets are possible. 
For instance, for rule 1 i can have the following set: 
1324 or even 3214 etc. leading to 4!. I am also capable of finding the total number of combinations which can have the sum equal to 10. Same for the second rule.

What is the answer?

kris19 · Answer

-------------------------------------------------------------------------------
 N is a four-digit number. N = ABCD where A, B, C, D are distinct single-digit whole numbers. How many possible values of N exist for any fixed set of four values of A, B, C, D?

(1) A + B + C + D = 10

if we assume, A = 1, B = 2, C = 3, and D = 4, then (1) is satisfied, and we get 4X3X2X1 = 24 four-digit numbers.
Now, we assume,A = 0, B = 1, C = 2, and D = 7, then (1) is satisfied, and we get 3x3x2x1 = 18 four -digit numbers
so, (1) alone is not sufficient.

(2) A + B + C - D = 8 
we can use the above approach to say (2) alone is not sufficient.

Now, (1) + (2), we get
A+B+C = 9, and D = 1
2+3+4 = 9, and D = 1, so we get 4x3x2x1 = 24 four-digit numbers
0+3+6 = 9, and D = 1, so we get 3x3x2x1 = 18 four-digit numbers,

so, we don't get a unique number of four-digit numbers, Answer is E.

MathRevolution · Answer

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 4 variables: Let the original condition in a DS question contain 4 variables. Now, we know that each condition (1) and (2) would usually give us an equation each, however, since we need 4 equations to match the numbers of variables and equations in the original condition, the unequal number of equations and variables should logically give us an answer E.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find possible values of N exist for any fixed set of four values of A, B, C, D.

=> N = ABCD

Second and the third step of Variable Approach: From the original condition, we have 4 variables (A, B, C, and D).To match the number of variables with the number of equations, we need 4 equations. Since conditions (1) and (2) will provide 1 equation each, E would most likely be the answer.

Let’s take a look at both conditions combined together.

Condition(1) tells us that A + B + C + D = 10.

Condition(2) tells us that A + B + C - D = 8.

=> A + B + C = 10 - D and A + B + C = 8 + D

=> 10 - D = 8 + D

=> D = 1 and hence A + B + C = 9

For A + B + C = 9: 2 + 3 + 4 = 9 = 4 * 3 * 2 * 1 = 24 four-digit numbers.

For A + B + C = 9: 0 + 3 + 6 = 9 = 3 * 3 * 2 * 1 = 18 four-digit numbers.

Since the answer is not unique, both conditions together are not sufficient by CMT 2.

So, E is the correct answer.

Answer: E

NOTE: When you know that the most likely answer is E, combined both the conditions and solve. Don't solve separate conditions.

N is a four digit number. N = ABCD where A, B, C, D are distinct

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GMAT Data Sufficiency (DS) Questions

N is a four digit number. N = ABCD where A, B, C, D are distinct

Prep Toolkit

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