The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,

Q: Why do we only consider cases starting with 4 and 5? What about 1, 2, or 3?

That is a fantastic question! If a four-digit number starts with 1, 2, or 3, it will always be less than 4321, regardless of what the other digits are. Since we are only interested in numbers strictly greater than 4321, we only need to focus on cases where the first digit is 4 or 5.

Q: Why do we multiply by 6 for the remaining positions?

Because the problem states that repetition of digits is allowed. We have 6 digits available: {0, 1, 2, 3, 4, 5}. For any position where there is no restriction, we have 6 independent choices.

Q: Why do we stop at the 4th digit in the final case?

We stop there because we have exhausted all the digits. In the final case, we have fixed the first three digits as 4, 3, and 2. The only remaining variable is the last digit, which must be chosen to satisfy the 'strictly greater than' condition. Once that is done, we have fully constructed the number.

Question

The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

Options:

(A) 288

(B) 306

(C) 360

(D) 310

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the number of four-digit numbers formed using the set $\{0, 1, 2, 3, 4, 5\}$ that are strictly greater than $4321$. Since repetition is allowed, each position can be filled by any of the $6$ available digits.

Note that a "four-digit number" implies the first digit cannot be $0$. However, in standard lexicographical counting problems of this type, we typically treat the positions as independent slots unless specified otherwise. We will proceed by partitioning the search space based on the thousands, hundreds, tens, and units digits.

### Case 1: The Freedom of the Five

If the thousands digit is $5$, the number is automatically greater than $4321$.

The remaining three positions can each be filled by any of the $6$ digits. The number of ways is:

$$1 	imes 6 	imes 6 	imes 6 = 216$$

### Case 2: The Four-Series

If the thousands digit is $4$, we must examine the hundreds digit. To be greater than $4321$, the hundreds digit must be greater than $3$.

Thus, the hundreds digit can be $4$ or $5$ ($2$ choices). The remaining two positions can be any of the $6$ digits. The number of ways is:

$$1 	imes 2 	imes 6 	imes 6 = 72$$

### Case 3: The Forty-Three Series

If the first two digits are $4$ and $3$, we examine the tens digit. To be greater than $4321$, the tens digit must be at least $2$.

If the tens digit is $3, 4,$ or $5$ ($3$ choices), the number is already greater than $4321$ regardless of the units digit. The units digit can be any of the $6$ digits. The number of ways is:

$$1 	imes 1 	imes 3 	imes 6 = 18$$

### Case 4: The Final Hurdle

If the first three digits are $4, 3,$ and $2$, we look at the units digit. To be strictly greater than $4321$, the units digit must be greater than $1$.

The valid choices for the units digit are $\{2, 3, 4, 5\}$. This gives us exactly $4$ possibilities.

### The Grand Total

By summing the results from all mutually exclusive cases, we arrive at the total count:

$$216 + 72 + 18 + 4 = 310$$

The total number of four-digit numbers formed from the set $\{0, 1, 2, 3, 4, 5\}$ that are strictly greater than $4321$ is **$310$**.

Animated Solution for Mathematics - Permutations and Combinations: The number of four-digit numbers strictly greater than 4321 that can be formed using the digits 0,1,2,3,4,5 (repetition of digits is allowed) is :

Select Answer:

Visualized Solution

Problem Setup

The Core Constraint

Case 1: Starting with 5

Case 1: Total Possibilities

Case 2: Starting with 4

Case 2: Total Possibilities

Case 3: Starting with 43

Case 3: Total Possibilities

Case 4: Starting with 432

Case 4: Total Possibilities

Total Number of Ways

The Sigma Insight: Fundamental Principle of Counting

Analyzing the Setup

Case 1

Case 2

Case 3

Case 4

The Grand Total

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