All The Five Digit Numbers In Which Each Successive Digit Exceeds Its GMAT Problem Solving

Question: All the five digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The 97th number in the list does not contains the digit

(A) 4
(B) 5
(C) 7
(D) 8
(E) 9

Correct Answer: (B)

Solutions and Explanation

Approach Solution - 1 :

Let us begin with the digit 12345. The unit digit can be raised from 5 to 9, which is 5 numbers. The following 5-digit number in the list would be 12356; by changing the unit digit from 6 to 9, we can make that number 4 numbers. Until the hundreds digit needs to be changed, we can continue doing this.

It is clear that the pattern sooner or later becomes 5 + 4 + 3 + 2 + 1 = 15 and that 12456 is the sixteenth number. Look at the final two digits 56 from the number 12356 after we increase the hundreds digit. In order to reach the following change in hundreds, 12567, we must add 4+3+2+1 = 10. Then, by repeating the pattern 15 + 10 + 6 + 3 + 1 = 35, we can reach 13456.

The first number where the thousands digit is changed is 13456. To do this, we have to add the digits 15 + 10 + 6 + 3 + 1 together. For each change in thousands, we can now repeat our earlier steps. For example, 14567 would have 35 + (10 + 6 + 3 + 1) = 55 numbers before it. The numbers before 15678 would be 55 + (6 + 3 + 1) = 65.

Replicate the numbers above, and then add 65 + (3 + 1) + 1 to get to 23456. We already completed the 20-number stretch from 13456 to 14567, so 20 more numbers are required to reach 24567. Following that, 24567 has 70+20 = 90 numbers ( which is 91 in the list) before it. Now that we can count up, we can discover that the 96th number is 24689 and 97th number is 24789.

Approach Solution - 2 :

The numbers would be arranged in a way that is given below.

12345
12346
12347
12348
12349
12356
12357
12358
12359 and will continue….

Numbers in total = 9C5 = 126. (every selection of 5 digits will give only 1 number)
Beginning with 1, 8c4 = 70 numbers

If we start with the number 2, then 7c4 = 35 numbers

Note that total of them 105 is greater than 97

Beginning with number 23, 6c3 = 20.
Total = 70 + 20 = 90 numbers

Beginning with 24, 5c3 = 10
Total = 90 + 10 = 100 > 97

Beginning with 24, the numbers would be arranged in a way that is given below.

24567
24568
24569
24678
24679
24689
24789 ---- 97th number

The digit 5 is missing

Approach Solution - 3 :

The five-digit numbers have the unique feature that each digit comes after the one before it. The first term in this series is 12345 because every digit in that number comes after the one before it. Beginning with number 1, we will count all five-digit numbers, moving on to numbers starting with number 2, and so forth until we reach the 97th number in the series. This will provide us with the needed answer.

The first group of possible numbers are those of the "1X" type. In this case, "X" stands for a four-digit number where each subsequent term is larger than the one before it.

The number of such “1X” type terms will be,

=> \(^8C_4\)
= 8! / (4! * 4!)
= 70

As a result, there are 70 terms of type "1X."
We will now determine how many numbers are of the type "2X."

=> \(^7C_4\)

= 7! / (4! * 3!)
= 35

The total number of terms is now 105 (70 + 35), which is higher than 97. As a result, our number's first term is 2.
The quantity of numbers of type "23Y," where "Y" stands for a three-digit number, which will be,

=> \(^6C_3\)
= 6! / (3! * 3!)
= 20

The quantity of terms of type "24Y" will be,

=> \(^5C_3\)
= 5! / (3! * 2!)
= 10

The total number of terms is now 100 (70+20+10), which is more than 97.

Consequently, 24 is the second term of our number. We can now simply continue by writing the following numbers in series.

Number - 91 is 24567
Number - 92 is 24568
Number - 93 is 24569
Number - 94 is 24678
Number - 95 is 24679
Number - 96 is 24689
Number - 97 is 24789

This makes it obvious that the 97th number in the series is 24678 and that it is lacking the digits 0, 1, 3, 5, and 9. Out of these digits, the digit 5 is one of the options and is therefore the answer.

“All the five digit numbers in which each successive digit exceeds” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

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