Question

Number of natural numbers less than 7000 using,0,1,3,7,9 (repetition allowed) is
A. 375
B. 275
C. 274
D. 374

Answer 1

Hint: When we have to calculate a natural number, we count from 1 as 0 is not a natural number. Also, we can use the following concept
 No. of ways to fill r places when repetition is allowed,
$\begin{align}
  & =n\times n\times n............r \\
 & ={{n}^{r}} \\
\end{align}$
First place can be filled in n ways, second, third, fourth … place can be filled in n ways as repetition is allowed.

Complete step-by-step answer:
We have to find the total no. of numbers that can be formed by using 0,1,3,7,9 and number is less than7000, so we have to consider the following case.
Case 1: One-digit number
i.e. filling of one place out of 0,1,3,7,9
This can be done by four ways as 0 is not a natural number
Case 2: Two-digits number
Tens place can be filled by 1,3,7,9 as 0 cannot be filled at tens place
i.e. by 4 ways and unit place can be filled by 0,1,3,7,9 i.e. 5 ways
so total number of ways to fill two places=$(4)(5)=20$
Case 3: Three-digit number
Hundred places can be filled by 1,3,7,9 i.e. 4 way as 0 cannot be filled at hundredth place, tenth and unit place can be filled by any of the digits 0,1,3,7, i.e.5 ways
So, Total number of three-digit number that can be formed by given digits$\Rightarrow (4)(5)(5)=100$
Case 4: Four-digit number
Thousand places can be filled by 1,3
i.e. by 2 ways as numbers is less than 7000 and 1000^th place cannot have 0 also 100th, 10^th and unit place can be filled by 0,1,3,7,9 i.e. 5 ways
so total number of ways to fill all the four places =$(2)(5)(5)(5)=250$
so total number of numbers that can be formed
$\begin{align}
  & =4+20+100+250 \\
 & =374 \\
\end{align}$
Hence option D is correct.

Note: This can be solved as 1000^th place filled by 3 ways, 100^th, 10^th and unit place by 5 ways each as repetition allowed. So, multiplication rule of country
$=3\times 5\times 5\times 5=375$
As 0 in not a natural number
So total number of natural numbers
$\begin{align}
  & =375-1 \\
 & =374 \\
\end{align}$
Also, it should be noted that If one operation can be performed in m ways, and (where it has been performed in any one of these ways) a second operation can then be performed in n ways; the number of ways of performing the two operations will be mn. If the two operations are mutually exclusive, that is, they are independent of each other then the number of ways of performing the two operations will be m+n.