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

Hint: 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 1^st 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 1^st 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.