How many natural numbers that are less than 10,000 can be GMAT Problem Solving

Question: How many natural numbers that are less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7, and 8?

A. 5000
B. 4096
C. 6560
D. 4095
E. 8000

Answer: D

Approach Solution (1):
We have to count all the numbers less than 10,000 which can be made with the digits 1, 2, 3, 0, 4, 5, 6 and 7.
We observe that the numbers less than 10,000 can have 4 digits, 3 digits, 2 digits and 1 digit. We will count all the possible numbers of each kind.
Firstly, we will consider the 1 digit numbers.
We can place any one of the 8 digits to form a one digit number.
Thus, the total number of 1 digit numbers which can be formed are 8.
We will now consider the 2 digit numbers.
We observe that we can place any one of the 8 digits at units place (Repetition allowed). However, 0 can’t be placed at tens place. So, we can have any one of the 7 numbers at the tens place.
Thus, the total number of 2 digit numbers which can be formed
= 8 × 7 = 56
We will now consider the 3 digit numbers.
We observe that we can place any one of the 8 digits at units and tens place (Repetition allowed). However, 0 can’t be placed at hundreds place. So, we can have any one of the 7 numbers at the hundreds place.
Thus, the total number of 3 digit numbers which can be formed
= 8 × 8 × 7 = 448
We will now consider the 4 digit numbers.
We observe that we can place any one of the 8 digits at units, tens and hundreds place (Repetition allowed). However, 0 can’t be placed at thousands place. So, we can have any one of the 7 numbers at the thousands place.
Thus, the total number of 4 digit numbers which can be formed
= 8 × 8 × 8 × 7 = 3584
So, the total numbers less than 10,000 are
= 3584 + 448 + 56 + 8 = 4096 – 1 = 4095
Hence, there are 4096 numbers less than 10,000 which can be formed from the given digits.
Correct option: D

Approach Solution (2):
We know that less than 10,000 means there can be 4 digits, 3 digits, 2 digits, 1 digit.
So now, the numbers can also be repeated.
Hence, Total number of Numbers formed = 8 * 8 * 8 * 8 = 4096 – 1 = 4095
Correct option: D

Approach Solution (3):
We are missing digits 5 and 9 in the question. so we have 8 digits for each position (ones, tens hundreds, thousands)
The numbers that can be constructed using 8 digits in 4 positions = (8 * 8 * 8 * 8) - 1 (To disregard the case 0000)
= 4096 - 1
= 4095
Correct option: D

“How many natural numbers that are less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7, and 8?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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