Question:
A five digit number divisible by $3$ is to be formed by using the numerals $0$, $1$, $2$, $3$, $4$ and $5$ without repetition. The total number of ways in which can be done is
A five digit number divisible by $3$ is to be formed by using the numerals $0$, $1$, $2$, $3$, $4$ and $5$ without repetition. The total number of ways in which can be done is
Updated On: Jun 14, 2022
- $216$
- $600$
- $240$
- $3125$
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The Correct Option is A
Solution and Explanation
Since, a five-digit number is formed using the digits $\{0,1,2,3,4$ and $5\}$ divisible by $3$ i.e. only possible when sum of the digits is multiple of three.
Case I Using digits $0, 1, 2, 4, 5$
Number of ways $= 4 \times 4 \times 3 \times 2 \times 1 = 96$
Case II Using digits $1, 2, 3, 4, 5 $
Number of ways $= 5 \times 4 \times 3 \times 2 \times 1 = 120$
$\therefore$ Total numbers formed $= 120 + 96 =216$
Case I Using digits $0, 1, 2, 4, 5$
Number of ways $= 4 \times 4 \times 3 \times 2 \times 1 = 96$
Case II Using digits $1, 2, 3, 4, 5 $
Number of ways $= 5 \times 4 \times 3 \times 2 \times 1 = 120$
$\therefore$ Total numbers formed $= 120 + 96 =216$
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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.