Given,
The numbers 0, 1, 1, 5, 9. Total of 5 digits, and it has 1 repeated digit 1 repeated twice.
To find : Number of numbers that can be formed using digits 0, 1, 1, 5, 9 in such a way that arranged number is greater than 50000.
Notice that an arrangement in which the first digit is either 5 or either 9 will only produce number greater than 50000. We have to find such numbers.
The problem can now be rephrased as to find a total number of permutations of 5 objects (0, 1, 1, 5, 9) of which two objects are of same type (1, 1), And all other objects are distinct.
But,
Either 5 or 9 will be only in the first place (According to question).
First,
We will find a total number of permutations of these 5 digits starting with 5 and then we will find the total number of permutations of these 5 digits starting with 9. Addition of these two numbers will give us the required numbers greater than 50000.
Total number of permutations starting with 5 will be equals to permutations of remaining digits (0, 1, 1, 9) in 4 remaining places = \(\frac{4!}{2!}\).
Total number of permutations starting with 9 will be equals to permutations of remaining digits (0, 1, 1, 5) in 4 remaining places = \(\frac{4!}{2!}\).
Number of permutations in which either 5 or 9 will come at first place will be equal to :
= \(\frac{4!}{2!}\) + \(\frac{4!}{2!}\)
= 24
Hence,
Total number of permutations of 5 digits (0, 1, 1, 5, 9) forming a 5 digit number greater than 50000 is equals to 24.
