Question

How many numbers between $2000$ and $5000$ can be made from the digits $1,2,4,5,7$ and $8$ if each digit is used only once?

Answer 1

We know that there are only two digits among the given numbers that could be in the thousandth position. We will fix each one of them in the leftmost position and try to find the number of numbers between $2000$ and $5000$ that can be made using the given digits without repetition.

Complete step by step solution:
Let us consider the given digits $1,2,4,5,7$ and $8.$
We need to find the number of numbers between $2000$ and $5000$ that could be made using the given digits.
We know that there are only two digits among the given digits that could be the leftmost digit which is the thousandth place. They are $2$ and $4.$ We can easily say that the numbers possessing $1,5,7$ and $8$ in the thousandth place do not belong to the required range. They are either less than $2000$ or greater than $5000.$
Let us fix $2$ in the thousandth place. Then there are $5$ digits that can be in the hundredth place. So, by default, we have $4$ digits in the tenth place and $3$ digits in the unit’s place.
So, in that case, we will get the number of numbers between $2000$ and $5000$ that start with $2$ by ${}^{5}{{P}_{3}}=5\times 4\times 3=60.$
Now let us fix $4$ in the thousandth place. Then there are $5$ digits that can be in the hundredth place. So, we have $4$ digits in the tenth place and $3$ digits in the unit’s place again. So, we will get ${}^{5}{{P}_{3}}=5\times 4\times 3=60$ numbers between $2000$ and $5000$ that start with $4.$
In total, we will get $60+60=120$ numbers between $2000$ and $5000$ using the given digits.
Hence there are $120$ numbers between $2000$ and $5000$ that can be made using the given digits.

Note: We know that we have used the permutation to find how many numbers we can make between $2000$ and $5000$ using the given digits. We should remember that the permutation is an arrangement of objects when the order matters.