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.
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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.