Question

The number of integers strictly lying between \[2\] lacs and \[8\] lacs which have at least \[2\] digits equal is less than
A. \[509280\;\]
B. \[509279\;\]
C. \[509281\;\]
D. \[509278\;\]

Answer 1

Hint: Here first we will find how many total numbers of integers lie in between \[2\] lacs and \[8\] lacs. After that we will take the condition that no two digits in those integers between \[2\] lacs and \[8\] lacs are equal to each other. Now therefore to find integers which have at least \[2\] digits equal we subtract the total number of integers and number of integers with no digits in them being the same.

Complete step by step answer:
Here subtracting, \[800000-200000 =600000\]. Therefore the total number of integers to find between \[2\] lacs and \[8\] lacs is \[6\] lacs. Now in this question we have been asked that we need to find the number of integers between \[2\] lacs and \[8\] lacs in the case that we have at least \[2\] digits equal.

Now to find this we must first find the number of integers in the case where there is no repetition allowed. We find this using permutation that is: let us solve this by taking cases where the \[5\] digits after the first are shown by “_”
Therefore:
\[2\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways
\[\Rightarrow 3\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways
\[\Rightarrow 4\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways
\[\Rightarrow 5\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways
\[\Rightarrow 6\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways
\[\Rightarrow 7\_\text{ }\_\text{ }\_\text{ }\_\text{ }\_=9\times 8\times 7\times 6\times 5\] Ways

Therefore total number of ways will be;
\[6\times \left( 9\times 8\times 7\times 6\times 5 \right) =90720\]
Now therefore this gives us the condition that no digits are repeated. If no digits are repeated there won’t be at least \[2\] same digits therefore subtraction the total number of digits from the case where no digits are repeated gives us our necessary answer
\[600000-90720 =509280\]
Therefore the number of integers strictly lying between \[2\] lacs and \[8\] lacs which have at least \[2\] digits equal is less than \[509280\].

Hence, the correct answer is option A.

Note:To understand more what permutation is just know that permutation is a way you can arrange any objects in a definite order. This is just a case where we need to find a solution only if there is replenishment. A question can also be formed where you just need the solution if there is no replacement and in that case you just need to solve the second part of the solution.