Question

Total numbers of 4 digit- numbers which are not divisible by $5$ are?
A. $7200$
B. $3600$
C. $14400$
D. $1800$

Answer 1

Hint: In order to find the total numbers of four digits that are not divisible by $5$, for that first we need to find the total numbers that are divisible by $5$ and subtract it from total four-digit numbers. In order to find the numbers that are divisible by $5$ use the permutation method of placing the values at the suitable places, making arrangements.

Complete step by step answer:
We are given the 4-digit numbers. So, we know the smallest four-digit number is $1000$ and the largest possible four-digit number is $9999$. So, the total number of 4- digit numbers are: $9999 - 1000 + 1 = 9000$

In order to calculate not divisible by $5$, we would first calculate the numbers that are divisible by $5$. We know that there are four spaces to fill the possible digits that can form a 4- digit number, so let’s place them. Since, from the divisibility rule of $5$, we know that a number should have $0$ or $5$ in the one’s place, so the one’s place of the four-digit can have two numbers that is only $0$ or $5$ and can be arranged in $2!$ Ways.

Next, for the first or thousandth term, we can place any digit between $0$ to $10$ except $0$ as it would make the term a 3-digit number again. So, the total numbers are $9$ and can be arranged in $9!$ Ways. The 2nd and 3rd terms that are hundredth and the tenth place can have any digits between $0$ to $10$, so the ways that $10$ digits can be arranged are $10!$.

Therefore, the total number of ways the four digits that are divisible by $5$ can be formed as: Product of ways at their respective positions i.e.
${\text{Total ways}} = 9 \times 10 \times 10 \times 2 = 1800$
Hence, the total four- digit numbers that are divisible by $5$ is $1800$. Since, there were total 4-digit numbers as $9000$ and the 4-digit numbers that are divisible by $5$ is $1800$.The difference between the two would give the total numbers that are not divisible by $5$. That is: total 4-digit numbers that are not divisible by $5$ is $9000 - 1800 = 7200$

Hence, option A is correct.

Note:$!$ is known as factorial, which multiplies every number from 1 to the number given. Permutation is used in this question because it was based on arrangements of four digits and order was important. And, we know that in permutation order is the most important factor.