Question

What is the number of natural numbers from 1000 to 9999 (both inclusive) that do not have all four different digits?

Answer 1

- Hint: Let us first know about four – digit numbers.
FOUR – DIGIT NUMBERS: The numbers that have four, i.e. 4 digits are called four – digit numbers. In other words, we can say that the numbers that have ones, tens, hundreds and thousands place are called four – digit numbers.
The formula that is used here is as follows
For choosing r different objects from n different objects, the formula that we can use is
\[={}^{n}{{C}_{r}}=\frac{n!}{r!\left( n-r \right)!}\]
We will also use the fundamental theorem which states that if n different events are to be considered at the same time, then we simply multiply the number of ways of each of the events.

Complete step-by-step solution -
Let us now solve the question.
We need to find the number of natural numbers from 1000 to 9999 (both inclusive) that do not have all four different digits
Total number of numbers from 1000 to 9999 = (9999 – 1000) + 1
                                                                                                           = 8999 + 1 = 9000
Numbers that have all four different digits = \[{}^{9}{{C}_{1}}\times {}^{9}{{C}_{1}}\times {}^{8}{{C}_{1}}\times {}^{7}{{C}_{1}}=9\times 9\times 8\times 7\] = 4536
(As for filling the first space of the number, we can select any number except 0. Then, for second place, we can select any number after any one is taken for the first place. Similarly, for the third place, we can select any number from the remaining 8 and for the fourth place; we can select any number from the remaining 7)
Number of numbers that do not have all 4 digits different = 9000 – 4536
                                                                                                            = 4464
Hence, the number of numbers that do not have all 4 digits different is 4464.
Note:- One should know the basic formulas of permutation and combination.
The students can make an error if they do not know the basic formulas of the permutation and combination.
And therefore, it would be very difficult for them to solve such questions if they do not know such formulas.