Question

How many numbers greater $40,000$ can be formed from the digits $2,4,5,5,7$
A) $12$
B) $24$
C) $36$
D) $48$

Answer 1

Hint: Initially fix the first 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 Solution
Given: Five digits $2,4,5,5,7$ are given by which we have to find how many numbers greater than $40,000$ can be formed.
Let us make a box of five sections to understand the concept easily.

1st

2nd

3rd

4th

5th

Initially, we have to fix a digit at 1st position and the first position can be filled by four digits out of $5$ digits (excluded $2$ ).
We can’t place $2$ at the first position because it is mandatory to form a no. greater then $40,000$and if we place $2$ at first position the no. will be less than $40,000$.
So 1st position can be filled in $ = 4$ ways.
The rest of the positions can be filled in $ = \dfrac{{4!}}{{2!}}$ ways.
We divided the outcome by $2!$ because$5$ comes two times in the number.
So the required ways in which numbers greater than can be formed from the digits $2,4,5,5,7$ are $ = $ The numbers of ways first position can be filled $ \times $ the no. of ways the rest of positions can be filled $ = 4 \times \dfrac{{4!}}{{2!}}$ways
$ = \dfrac{{4 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}}$
$48$ ways.

Hence $48$ numbers greater than $40,000$can be formed from the digits $2,4,5,5,7$.

Note:
After finding out the possibilities of the first position we will apply the concept of permutation and will divide by $2!$ because $5$ comes two times and if a no. comes $3$ times then it must be divided by $3!$ and so on.