How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition) GMAT Problem-Solving

Question: How many five digit numbers can be formed from 1, 2, 3, 4, 5 (without repetition), when the digit at the unit’s place must be greater than that in the ten’s place?

A. 54
B. 60
C. 17
D. 2×4!
E. 120

Answer: B

Solution and Explanation:

Approach Solution 1:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Without repeating digits, the total number of numbers that may be created from the digits 1, 2, 3, and 5 is 5*4*3*2*!, which equals 5! = 120.
Now, the unit digit will be larger than the tenth digit in half of them and smaller in the other half.
Let's take the example of the numbers 1, 2, and 3. 3*2*1=6 is the total number of integers without repeating digits.
Numerals whose unit digit is higher than the tenth digit
123, 213, 312
Numbers whose tenth digit is higher than the digit of the unit
321, 132, 231
Total Number of Cases = 120/2 = 60.
Hence,
Correct option: B

Approach Solution 2:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
Ten's place > unit's place
Therefore, a hypothetical unit digit would be = 2.3.4.5
When 2 is in units, the first digit must be in tens, while the other numbers are formed by (3,4,5).
number of possibilities = 3!= 6
Similar to how 1 or 2 might be in the tenth digit when 3 is in the unit digit, and 3 more digits make up the number.
so the final number is 3!*2=12.
After 4 is reached once more, the total number of outcomes is 3!*3=18.
and when 5, the total number that could occur is 3!*4=24.
total possible outcomes = 6 + 12 + 18 + 24 = 60
Correct option: B

Approach Solution 3:
Apply the information in the question to the GMAT question at hand. These problems apply to numerous disciplines of mathematics. This question has to do with permutations and combinations. It is challenging to select the best option because of the way the options are presented. Candidates must be able to comprehend the appropriate approach to eliciting the desired response. Out of the five possible answers, there is only one that is correct.
The sum of all five-digit numbers is five, which equals 120.
By symmetry, the units digit will be greater than the tens digit in half of them and the tens digit will be greater than the units digit in the other half.
So 120/2 = 60
Answer (B)
Noting the symmetry, all such numbers will not be included if 1 is in the units digit. All such numbers will be included if the units digit is 5, as in this case. Only numbers with 1 in the tens digit will be included if 2 is in the units digit. Only numbers with 5 in the tens digit will be excluded if 4 is in the units digit. When 3 appears in the units digit, 50% of the numbers will be valid and 50% won't.
Correct option: B

“How many numbers can be formed from 1, 2, 3, 4, 5 (without repetition)" - 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.

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