Question

The number of numbers from $1000$ to $9999$ (both inclusive) that do not have all $4$ different digits, is
A) 4048
B) 4464
C) 4518
D) 4536

Answer 1

Hint: This is a very interesting question having formation of numbers using digits. Here we have to first check all the possibilities. At first we will check possibilities for the first place digit. It can be considered by the permutation and combination when terms are repeated. Then find the possibilities when the terms are not repeated. Subtract both the values to get the final result.

Complete step-by-step answer:
We have to find the possibilities of all $4$ different digits.
So we can say that
Total number of numbers $ - $ all $4$ different digits
So we can wright total number of digits $ = $ 1000 to 9999
Here we clearly see that the possibilities to fill the first position of the number is $1.2.3.....9$ then there total $9$ possibilities to fill the first.
Now the possibilities to fill the second number is $0.1.2.3.....9$ so the possibilities to fill this is $10$.
Similarly the third and fourth number also filled with possibilities of $10$. (Because here number can repeat)
Total number: -
$
  9 \times 10 \times 10 \times 10 \\
   \Rightarrow 9000 \\
 $
All $4$ different digits
Here to fill the first number we let a number that is $3$
Then the possibilities to fill the second place $0.1.2.4.....9$ the total number of possibilities is $9$$\left( {{\text{3 already fixed}}} \right)$.
Similarly third and fourth numbers possibility to fill is$8$,$7$
So, total number: -
$ \Rightarrow $$9 \times 9 \times 8 \times 7 = 4536$
Now required answer will be
$ \Rightarrow 9000 - 4536 = 4464$
$\therefore $ The required number of numbers is 4464.

Thus option (2) is the correct answer.

Note: Here we have to always keep in mind that in non-repeated cases when we are going to check for a number of possibilities we have to decrease the total number by 1 every time and multiply them. This problem is a good example of Permutation and combination. These two are related to certain differences. Their computations in different situations are being done by using suitable arrangements/ selections and formulas. Permutations and combinations are the numbers of different ways in which some given objects from a set may be selected, generally without the replacement. Hence it will give a subset. This selection of subsets is termed as permutation while the order of selection is important, combination while the order is not significant at all.