Question: Let abcd be a general four-digit number and all the digits are non-zero. How many four-digits numbers abcd exist such that the four digits are all distinct and such that a + b + c = d?

(A) 6
(B) 7
(C) 24
(D) 36
(E) 42

Correct Answer: E
Solution and Explanation:

Approach Solution 1:

This is a GMAT problem-solving question in which you have to use the details given in the question to solve the problem. The problem in this category are coming from different areas of math topics. This one particularly is from number theory.
The option is given in such a way that it is difficult to guess the correct answer. The candidates need to know the right approach to get the required answer. Only one of the given five choices is correct.

The smallest possible number for the equation abcd is 6, while the highest is also 6.
1+2+x:

The least possible value of x is 3, while the maximum is 6.
6−3+1=4

There are four values for x, and when they are multiplied by 1 and 2, they produce a digit.
NUMBERING 4
1+3+x:
X can only be either 4 or 5. 2 was already discussed in the prior section. We get 8 if x is 4 and 9 if x is 5.

NUMBER 2
2+3+x:
2 + 3 = 5, so 1 x 4 is the answer. One has already been covered, two and three won't work because they wouldn't be distinct, leaving only number four.

NUMBER 1
Total: 4+2+1=7
Since 4 + 2 + 1 = 7, there are a total of 7 numbers that are valid. However, there are 3! 3! many ways to arrange the three numerals (abc) before the d.
7∗3! = 7∗(3*2*1) = 42

E is the correct answer.

Approach Solution 2:

The problem statement states that:

Given:

• Let abcd be a general four-digit number and all the digits are non-zero.

Asked:

• How many four-digits numbers abcd exist such that the four digits are all distinct and such that a + b + c = d

There are 7 combinations:
abcd
1236
2349
1359
1348
1247
1258
1269
Each can be rearranged in 6 ways.
Therefore, Total four-digits numbers abcd = 6*7 = 42 .

Approach Solution 3:

The problem statement informs that:

Given:

• Let abcd be a general four-digit number and all the digits are non-zero.

Asked:

• How many four-digits numbers abcd exist such that the four digits are all distinct and such that a + b + c = d

There are 3!=6 ordered triples (a,b,c) for a given set , since a,b,c are distinct. The set uniquely determines d provided a+b+c ≤ 9.

We count the number of sets {a,b,c} with 1 ≤ a < b < c. Then c ∈ {3,4,5,6} (lower bound is obvious, the upper bound follows from the fact that c ≥ 7 implies d=a+b+c ≥ 7+2+1).

If c=3, then a=1 and b=2. There is only one such set.
If c=4, then a, b must be chosen from the set {1,2,3} and a+b+c ≤ 2+3+4 ≤ 9. There are \(\binom{3}{2}\)such sets.
If c=5, then a, b must be chosen from the set {1,2,3,4}, but since a+b ≤ 4, there are only two such sets.
If c=6, then a, b must be chosen from the set {1,2,3,4,5}, but since a+b ≤ 3, there is only one such set.

Therefore, there are (1+3+2+1) × 3! =42 such numbers

“Let abcd be a general four-digit number and all the digits are non-zero" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”. To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.The candidates can practice GMAT Quant practice papers to improve their knowledge of mathematics.

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