Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1,2 , 3 and 5 , and are divisible by 15 , is equal to _____
Correct Answer: 21
Approach Solution - 1
Determining Numbers Divisible by 15
To determine numbers divisible by 15, the following conditions must be met:
- The last digit must be 5 (a condition for divisibility by 5).
- The sum of the digits must be divisible by 3 (a condition for divisibility by 3).
Possible Combinations
| Combination | Numbers Formed | Sum of Digits |
|---|---|---|
| 1215 | 3 | 9 |
| 2235 | 3 | 12 |
| 3115 | 3 | 10 |
| 1155 | 3 | 12 |
| 2355 | 6 | 15 |
| 3555 | 3 | 18 |
Explanation:
For each combination of digits, the number of possible arrangements (permutations) that satisfy the conditions is determined. For example, the combination 1215 has 3 valid numbers since the arrangement must end with 5 and satisfy divisibility by 3.
Similar calculations are performed for all other combinations.
Total Numbers:
Total Numbers = 3 + 3 + 3 + 3 + 6 + 3 = 21
Approach Solution -2
For number to be divisible by 15, last digit should be 5 and sum of digits must be divisible by 3.
Top Questions on Combinations
- A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?
- The number of ways in which a committee of 7 members can be formed from 6 teachers, 5 fathers and 4 students in such a way that at least one from each group is included and teachers form the majority among them, is:
- AP EAPCET - 2025
- Mathematics
- Combinations
- A group of 9 students, s1, s2,…., s9, is to be divided to form three teams X, Y and Z of sizes 2, 3, and 4, respectively. Suppose that s1 cannot be selected for the team X and s2 cannot be selected for the team Y. Then the number of ways to form such teams, is _______.
- JEE Advanced - 2024
- Mathematics
- Combinations
The value of 49C3 + 48C3 + 47C3 + 46C3 + 45C3 + 45C4 is:
- KCET - 2024
- Mathematics
- Combinations
- A committee of 11 members is to be formed out of 8 males and 5 females. If \( m \) is the number of ways the committee is formed with at least 6 males and \( n \) is the number of ways with at least 3 females, then:
- MHT CET - 2024
- Mathematics
- Combinations
Questions Asked in JEE Main exam
Which one of the following graphs accurately represents the plot of partial pressure of CS₂ vs its mole fraction in a mixture of acetone and CS₂ at constant temperature?
- JEE Main - 2026
- Organic Chemistry
- For two identical cells each having emf \(E\) and internal resistance \(r\), the current through an external resistor of \(6\,\Omega\) is the same when the cells are connected in series as well as in parallel. The value of the internal resistance \(r\) is ________ \(\Omega\).
- JEE Main - 2026
- Current electricity
- For three unit vectors \( \vec a, \vec b, \vec c \) satisfying \[ |\vec a-\vec b|^2 + |\vec b-\vec c|^2 + |\vec c-\vec a|^2 = 9 \] and \[ |2\vec a + k\vec b + k\vec c| = 3, \] the positive value of \( k \) is:
- JEE Main - 2026
- Vector Algebra
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$)
- JEE Main - 2026
- Rotational Mechanics
- The system of linear equations
$x + y + z = 6$
$2x + 5y + az = 36$
$x + 2y + 3z = b$
has- JEE Main - 2026
- Matrices and Determinants
Concepts Used:
Permutations and Combinations
Permutation:
Permutation is the method or the act of arranging members of a set into an order or a sequence.
- In the process of rearranging the numbers, subsets of sets are created to determine all possible arrangement sequences of a single data point.
- A permutation is used in many events of daily life. It is used for a list of data where the data order matters.
Combination:
Combination is the method of forming subsets by selecting data from a larger set in a way that the selection order does not matter.
- Combination refers to the combination of about n things taken k at a time without any repetition.
- The combination is used for a group of data where the order of data does not matter.