Question

How many natural numbers less than $1000$ are there in which no two digits are repeated?
        $a.{\text{ 738}}$
        $b.{\text{ 792}}$
        $c.{\text{ 837}}$
        $d.{\text{ 720}}$

Answer 1

Hint: First we’ll see how many types of numbers are less than 1000, one digit, two digit, three digit and then we'll see how many choices we have to fill their number places.

We know that,
Numbers less than $1000$ are one digit numbers, two digit numbers and three digit numbers.
Now, number of one digit number$ = 9$
Similarly, number of two digit numbers in which no two digits are repeated$ = 9*9 = 81$
 (for two digit number, for one’s place we have$9$ choices, for ten’s place we have $9$ choices ($0$excluded))
Similarly, for three digit numbers
Number of three digit number in which no two numbers are repeated
$\begin{gathered}
   = 9*9*8 \\
   = 648 \\
\end{gathered} $
\[\therefore \]Total natural numbers less than $1000$ in which no two digits are repeated
$\begin{gathered}
   = 9 + 81 + 648 \\
   = 738 \\
\end{gathered} $
Therefore, the correct answer is option $a$.

Note: There are total 999 natural numbers under 1000 in which we have 261 no.’s where 2 digits are repeated which includes numbers like 11,22,33,44,100, 101, 110,200, 202,220 etc. making 738 (\[\because \]999-261) numbers with no repeated digits.