Question

How many numbers can be made with the help of the digits $0,1,2,3,4,5$ which are greater than $3000$ (repetition is not allowed)?
A. 180
B. 360
C. 1380
D. 1500

Answer 1

Hint: First, we shall analyze the given data so that it can be easy for us to solve the problem. We are given six digits $0,1,2,3,4,5$ that can be used to make the numbers that are greater than $3000$. Follow the given step-by-step solution to solve the problem.

Complete step by step answer:
The given digits are $0,1,2,3,4,5$
Since we need to find the numbers that are greater than $3000$, the first digit cannot be the digit $0,1,2$
Also, there can four-digit numbers, five-digit numbers, and six digits numbers that are greater than $3000$
First, we shall consider the four-digit numbers.
 Since we need to find the numbers that are greater than $3000$, the first digit cannot be the digit $0,1,2$
Hence, the thousands can be filled by the digits $3,4,5$ .
That is there are three ways to fill the thousands.
Since repetition is not allowed, the hundreds can be filled by the remaining digits $0,1,2,4,5$
That is there are five ways to fill the hundreds.
Similarly, the tens can be filled by the remaining four digits. That is there are four ways to fill the tens.
Similarly, the ones can be filled by the remaining three digits, (i.e.)there are three ways to fill the ones.
Hence, number of four-digit numbers$ = 3 \times 5 \times 4 \times 3$
                                                                    $ = 180$
Next, we shall consider the five-digit numbers.
 Since we need to find the numbers that are greater than $3000$, the first digit cannot be the digit $0$
Hence, the ten thousand can be filled by the digits $1,2,3,4,5$ .
That is there are five ways to fill the ten thousand.
Since repetition is not allowed, the thousands can be filled by the remaining digits $0,1,2,3,4,5$
That is there are five ways to fill the thousands.
Similarly, the hundreds can be filled by the remaining four digits. That is there are four ways to fill the hundreds.
Similarly, the tens can be filled by the remaining three digits. That is there are three ways to fill the tens.
Similarly, the ones can be filled by the remaining two digits, (i.e.) there are two ways to fill the ones.
Hence, number of five-digit numbers $ = 5 \times 5 \times 4 \times 3 \times 2$
                                                                   $ = 600$
Next, we shall consider the six-digit numbers.
 Since we need to find the numbers that are greater than $3000$, the first digit cannot be the digit $0$
Hence, the lakhs can be filled by the digits $1,2,3,4,5$ .
That is there are five ways to fill the lakhs.
Since repetition is not allowed, the ten thousand can be filled by the remaining digits $0,1,2,3,4,5$
That is there are five ways to fill the ten thousand.
Similarly, the thousands can be filled by the remaining four digits. That is there are four ways to fill the thousands.
Similarly, the hundreds can be filled by the remaining three digits. That is there are three ways to fill the hundreds.
Similarly, the tens place can be filled by the remaining two digits. That is there are two ways to fill the tens.
Similarly, the ones can be filled by the remaining one digit, (i.e.) there is one way to fill the ones.
Hence, number of six-digit numbers$ = 5 \times 5 \times 4 \times 3 \times 2 \times 1$
                                                               $ = 600$
Hence, total numbers can be made$ = 180 + 600 + 600$
                                                                $ = 1380$

So, the correct answer is “Option C”.

Note: For constructing the four-digit number, the first digit cannot be the digit $0,1,2$ since we need to find the numbers that are greater than $3000$
            For constructing the five-digit numbers and six-digit numbers, the first digit cannot be the digit $0$.