How Many Natural Numbers Not Exceeding 4321 can be Formed GMAT Problem Solving

Question: How many natural numbers not exceeding 4321 can be formed with the digits 1,2,3,4 if the digits can repeat?

1. 64
2. 313
3. 4^4* 4
4. 256
5. 4^4/4!

Solutions and Explanation
Approach Solution : 1

Scenario - 1 : A four-digit number
=> 4*4*4*4 = 256
There are 256 possible ways to form a four-digit number in total.
The amount of ways that the four-digit numbers that are higher than 4321 can be formed are as follows:
Let's say the hundreds digit is either 3 or 4 and the thousand's digit is 4.
Therefore the total ways = 2*4*4 = 32
However, 4311, 4312, 43113, 43114, and 4321 are all less than or equal to 421.
The remaining ways are 256 - (32-5) = 229
Scenario - 2 : A three-digit number
There are four ways to fill in the hundredth digit.
The tens digit and the units digit can both be filled in four different ways each.
This is due to the fact that digit repetition is permitted.
Therefore total three-digit number = 4*4*4 = 64
Scenario - 3: A two-digit number
There are four different ways to fill in both the unit and tens digits. This is due to the fact that digit repetition is permitted.
Therefore total two-digit numbers = 4*4 = 16.
Scenario - 4 : A single-digit number
One digit numbers are limited to four.
The numbers needed are, 229+64+16+4 = 313

Correct Answer: (B)

Approach Solution : 2

Since there are 4 digits, 1, 2, 3, and 4, it is acceptable to repeat digits.
4 total 1-digit numbers are present.
4*4 = 16 is the total number of 2-digit numbers.
4*4*4 = 64 is the total number of 3-digit numbers
64 is the total number of 4-digit numbers starting with the number 1. (The number one position is occupied by 1)
4*4*4 = 64 is the total number of 4-digit numbers that start with the number 2.
4*4*4 = 64 is the total number of 4-digit numbers that start with the number 3.
4 * 4 = 16 is the total number of 4-digit numbers starting with 41.
4 x 4 = 16 is the total number of 4-digit numbers that start with 42.
4 is the number of all 4-digit numbers that start with 431
1 is the number of 4-digit numbers that start with 432 (4321 only) .
The number of total 4-digit numbers is 229, which is calculated as follows, 64 + 64 + 64 + 16 + 16 + 4 + 1 = 229.
Therefore, the sum of all natural numbers up to and including 4321 is = to 4 + 16 + 64 + 229 = 313.

Correct Answer: (B)

“Permutations and Combinations” - is a subject that the GMAT quantitative reasoning section covers. A school needs to be proficient in a variety of qualitative skills to answer the GMAT Problem Solving questions correctly. There are 31 questions in the GMAT Quant section as a whole. The GMAT Quant topics' problem-solving section calls for the resolution of calculative mathematical puzzles.

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