Question

How many numbers greater than $40,000$ can be formed by using the digits $1,2,3,4$ and $5$. If each is used only once in a number?
1. 24
2. 78
3. 32
4. 48

Answer 1

Initially fix the position with $4$ or greater than $4$ to make a number greater than $\,40,000$ and solve the rest of the part by the method of permutation to find the number of ways.

Complete step by step answer:
Given: Five digits $\,1,2,3,4$ and $5$ are given by which we have to find how many numbers greater than$\,40,000$ can be formed.
Make a box of the five section to understand the concept easily

Initially, we have to fix a digit at 1st position and first position can be filled by two digits out of 5 digits (excluded $1,2,3$)
We can’t place $1\,or\,2\,or\,3$ at the first position because it is mandatory to form a no. greater than $\,40,000$.
So 1st position can be filled $=2$ ways.
Formula for permutation $={}^{n}{{p}_{r}}$
The rest of the positions can be filled in $={}^{4}{{p}_{4}}=4!$ ways.
The number of ways first position $\,\,\times$ the no. of ways the rest of positions can be filled
$=\,2\times {}^{4}{{p}_{4}}$
$=2\times 4!$
Further simplifying we get:
$=2\times 4\times 3\times 2\times 1$
$=48\,$Ways.
Hence$\,48\,$numbers greater than$\,40,000$ can be formed from the digits$\,1,2,3,4$ and$\,5$.
So, the correct answer is “Option 4”.

Note: After finding out the possibility of the first position we will apply the concept of permutation. If the question has $\,5$ comes two times then we will divide by $2!$ with possible ways in the rest of the positions and if a no. comes $\,3$ times then it must be divided by $3!$ and so on. So, in this way we can solve similar types of problems.