Question

how many numbers are between 400 and 1000 can be formed with the digit 2,3,4,5,6,0.

Answer 1

if 0 is place at the first place then it will be of two digit number which is less than 400.

Answer 2

$\underline{\underline{\Huge{\textbf{Question:}}}}$
How many numbers are between 400 and 1000 can be formed with the digit 2,3,4,5,6,0?



$\underline{\underline{\Huge{\textbf{Solution:}}}}$

Given,
$\sf\textsf{Allowed Digits to form the Number = 2, 3, 4, 5, 6, 0}$

$\sf\textsf{Range within which the No. formed should fall \in (400 , 1000)}$

$\sf\textsf{Since the Number has to be in the Range \in (400 , 1000)}$
$\sf\textsf{The Number to be formed is a \textbf{3 - digit Number > 400}}$

$\begin{array}{|c|c|c|}\cline{1-3} Hundred's&Ten's& Unit's\\Place&Place&Place\\\cline{1-3}\end{array}$

$\sf\textsf{Here,}$
$\sf\textsf{There are \large{\textbf{2 Cases}}}$

$\begin{aligned}(1) & \bf No\: Repetition\: of\: the\: digits\: is\: allowed \\(2) & \bf Repetition\: of\: the\: digits\: is\: allowed\end{aligned}$



$\underline{\Huge{\textsf{Case-1:}}}$
$\sf\textsf{Assuming No. Digit Repetition is Allowed}$

$\sf\textsf{If the Repetition is not allowed,}\\ \sf\textsf{Total No. of allowed digits after use of}\\\sf\textsf{each digits from the available digits has to be noted.}$

$\sf\textsf{Total No. of Allowed digits = 6}$

$\underline{\large{\textsf{Step-1:}}}$
$\sf\textsf{Find the No. of ways of Filling the Hundreds Place}$

$\sf\textsf{The Hundred's Place of the Number cannot be 0, 2, 3}$

$\sf\textsf{The Hundreds Place can be Filled by 4, 5, 6}$
$\sf\textsf{No. of ways of Filling the Hundreds Place = 3}$

$\sf\textsf{One number is used, so total available digits = 5}$



$\underline{\large{\textsf{Step-2:}}}$
$\sf\textsf{Find the No. of ways of Filling the Ten's Place}$

$\sf\textsf{The Ten's Place of the Number}\\\sf\textsf{cannot be the number}\\\sf\textsf{used for Hundred's digit}$

$\sf\textsf{The Ten's Place can be Filled by using the available 5 digits }$

$\sf\textsf{No. of ways of Filling the Tens Place = 5}$

$\sf\textsf{Two numbers are used, so total available digits = 4}$



$\underline{\large{\textsf{Step-3:}}}$
$\sf\textsf{Find the No. of ways of Filling the Unit's Place}$

$\sf\textsf{The Unit's Place of the Number}\\\sf\textsf{cannot be the number used for}\\ \sf\textsf{Hundred's digit and Ten's digit}$

$\sf\textsf{The Unit's Place can be Filled by using the available 4 digits }$

$\sf\textsf{No. of ways of Filling the Unit's Place = 4}$



$\underline{\large{\textsf{Step-4:}}}$
$\sf\textsf{Find the No. of Numbers between 400 and 100}\\\sf\textsf{that can be formed with digits 2, 3, 4, 5, 6, 0}\\\textsf{when repetition of digits is not allowed}$

$\textbf{No. of Numbers that can be formed}\\ \textbf{is Product of No. of ways of} \\ \textbf{Filling the Hundred's, Ten's and Unit's Place}$

$\therefore \sf\textsf{No. of Numbers that can be formed \textbf{without repetition} = 3 \times 5 \times 4 = 60}$



$\underline{\Huge{\textsf{Case-2:}}}$
$\sf\textsf{Assuming the Digit Repetition is Allowed}$

$\sf\textsf{Total No. of Allowed digits = 6}$

$\underline{\large{\textsf{Step-1:}}}$
$\sf\textsf{Find the No. of ways of Filling the Hundreds Place}$

$\sf\textsf{The Hundred's Place of the Number cannot be 0, 2, 3}$

$\sf\textsf{The Hundreds Place can be Filled by 4, 5, 6}$
$\sf\textsf{No. of ways of Filling the Hundreds Place = 3}$



$\underline{\large{\textsf{Step-2:}}}$
$\sf\textsf{Find the No. of ways of Filling the tens Place}$

$\sf\textsf{The Ten's Place can be Filled by 2, 3, 4, 5, 6, 0}$

$\sf\textsf{No. of ways of Filling the Ten's Place = 6}$



$\underline{\large{\textsf{Step-3:}}}$
$\sf\textsf{Find the No. of ways of Filling the Unit's Place}$

$\sf\textsf{The Unit's Place can be Filled by 2, 3, 4, 5, 6, 0}$

$\sf\textsf{No. of ways of Filling the Unit's Place = 6}$



$\underline{\large{\textsf{Step-4:}}}$
$\sf\textsf{Find the No. of Numbers between 400 and 100}\\\sf\textsf{that can be formed with digits 2, 3, 4, 5, 6, 0}\\\textsf{when repetition of digits is allowed}$

$\textbf{No. of Numbers that can be formed}\\ \textbf{is Product of No. of ways of} \\ \textbf{Filling the Hundred's, Ten's and Unit's Place}$

$\therefore \sf\textsf{No. of Numbers that can be formed \textbf{with repetition} = 3 \times 6 \times 6 = 108}$


$\therefore$

$\blacksquare\: \: \sf\textsf{No. of Numbers that can be formed \textbf{without repetition} =\underline{\Large{\textbf{60}}}}$

$\blacksquare\: \: \sf\textsf{No. of Numbers that can be formed \textbf{with repetition} = \underline{\Large{\textbf{108}}}}$