The number of numbers between 2,000 and 5,000 that can be formed with the digits 0,1,2,3,4 (repetiti

Q: Why can't I just use the permutation formula $P(5, 4)$?

That is a classic trap! The permutation formula assumes all arrangements are valid. Here, we have two major constraints: the thousands place cannot be $0$ (or $1$ or $5$), and the number must be divisible by $3$. You must filter your sets first, then apply permutations.

Q: Why do we drop a digit?

We have $5$ digits available $\{0, 1, 2, 3, 4\}$, but we need a $4$-digit number. The sum of all $5$ digits is $10$. By dropping one digit $x$, the sum of the remaining $4$ digits becomes $10-x$. We choose $x$ such that $10-x$ is a multiple of $3$.

Q: Does the order of digits matter when checking divisibility?

No! That is the beauty of the divisibility rule for $3$. The sum of the digits is invariant under permutation. Whether you write $1230$ or $3210$, the sum is $6$, so both are divisible by $3$. We only worry about the order when we calculate the number of arrangements.

Question

The number of numbers between 2,000 and 5,000 that can be formed with the digits 0,1,2,3,4 (repetition of digits is not allowed) and are multiple of 3 is :-

Options:

(A) 36

(B) 48

(C) 24

(D) 30

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with finding the number of integers between $2000$ and $5000$ that are multiples of $3$, using only the digits $\{0, 1, 2, 3, 4\}$. This problem requires balancing a range constraint with a divisibility constraint.

The range $2000 < N < 5000$ implies that the number must have exactly $4$ digits. The thousands place digit, let us call it $d_1$, must be chosen from the set $\{2, 3, 4\}$.

### The Divisibility Secret

A number is a multiple of $3$ if and only if the sum of its digits is a multiple of $3$. The available digits are $\{0, 1, 2, 3, 4\}$, which have a total sum of $10$.

Since we are forming a $4$-digit number, we must exclude exactly one digit, $x$, from the set. The sum of the remaining digits will be $10 - x$. For this sum to be a multiple of $3$, $10 - x$ must be $9$ or $6$.

Testing the possibilities for $x$:
*   If $x = 1$, the sum is $9$ (Valid).
*   If $x = 4$, the sum is $6$ (Valid).

This leaves us with two distinct sets of digits to form our numbers:
Set A: $\{0, 1, 2, 3\}$
Set B: $\{0, 2, 3, 4\}$

### Case Analysis

**Case 1: Using the set $\{0, 1, 2, 3\}$**
The thousands place $d_1$ must be from $\{2, 3\}$ because $0$ cannot be in the thousands place and $1$ is allowed but does not restrict the range. There are $2$ choices for $d_1$.

The remaining $3$ positions can be filled by the remaining $3$ digits in $3!$ ways.
$$Total_1 = 2 	imes 3! = 2 	imes 6 = 12$$

**Case 2: Using the set $\{0, 2, 3, 4\}$**
The thousands place $d_1$ can be any of $\{2, 3, 4\}$. There are $3$ choices for $d_1$.

The remaining $3$ positions can be filled by the remaining $3$ digits in $3!$ ways.
$$Total_2 = 3 	imes 3! = 3 	imes 6 = 18$$

### Final Calculation

Since these two cases are mutually exclusive, we sum the results to find the total count of valid integers.

$$Total = 12 + 18 = 30$$

**The total number of such integers is $30$.**

Animated Solution for Mathematics - Permutations and Combinations: The number of numbers between 2,000 and 5,000 that can be formed with the digits 0,1,2,3,4 (repetition of digits is not allowed) and are multiple of 3 is :-

Select Answer:

Visualized Solution

The $4$ -Digit Constraint

Thousands Place Restriction

Divisibility by $3$

Sum of Available Digits

Choosing the Digit to Drop

Finding Valid Sets

Analyzing Set 2: ${0, 1, 2, 3}$

Permutations for Set 2

Analyzing Set 1: ${0, 2, 3, 4}$

Permutations for Set 1

Final Calculation

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Divisibility Secret

Case Analysis

Final Calculation

Similar Questions

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

The total number of 3 -digit numbers, whose sum of digits is 10, is

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Animated Solution for Mathematics - Permutations and Combinations: The number of numbers between 2,000 and 5,000 that can be formed with the digits 0,1,2,3,4 (repetition of digits is not allowed) and are multiple of 3 is :-

Select Answer:

Visualized Solution

The .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 4-Digit Constraint

Thousands Place Restriction

Divisibility by .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 3

Sum of Available Digits

Choosing the Digit to Drop

Finding Valid Sets

Analyzing Set 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } {0,1,2,3}

Permutations for Set 2

Analyzing Set 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } {0,2,3,4}

Permutations for Set 1

Final Calculation

The Sigma Insight: Combinations and Selection

Analyzing the Setup

The Divisibility Secret

Case Analysis

Final Calculation

Similar Questions

The total number of three-digit numbers, with one digit repeated exactly two times, is ______.

The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _______.

The total number of 3 -digit numbers, whose sum of digits is 10, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of functions f from {1,2,3,…,20} onto {1,2,3,…,20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is:

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S={1,2,3,5,7,10,11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is ____.

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of matrices of order 3×3, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is

The $4$ -Digit Constraint

Divisibility by $3$

Analyzing Set 2: ${0, 1, 2, 3}$

Analyzing Set 1: ${0, 2, 3, 4}$

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