Question

how many three digit odd numbers can be formed from the digits 1,2,3,4,5 and 6 when
(a) repetition is not allowed
(b) repetition is allowed; how many three digit odd numbers can be formed from the digits 1,2,3,4,5 and 6 when; (a) repetition is not allowed; (b) repetition is allowed

Answer 1

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

$\textsf{Given,}\\\textsf{Allowed digits = 1, 2, 3, 4, 5}$

To form 3-digit Odd Numbers.

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

$\underline{\Huge{\textsf{Case-a}}}$
$\underline{\textit{Repetition is allowed.}}$

In this case, it is necessary to track
Total Numbers remained in allowed digits after filling each digit.

$\underline{\LARGE{\textsf{Step-1}}}$
$\textsf{Find the Number of ways of Filling}\\\textsf{the Unit's Place}$

As the Number to be formed must be a odd number, the unit digit of the number must be an odd number from the allowed digits.

Odd No.s from the Allowed digits = 1, 3, 5

$\square\quad\square\quad \! \! \! \underbrace{\! \! \blacksquare\! \!}_{1/2/3}$

No. of ways of ways of filling the Unit's place = 3

Remaining Allowed Digits = 5

$\underline{\LARGE{\textsf{Step-2}}}$
$\textsf{Find the Number of ways of Filling the}\\\textsf{Hundred's Place.}$

As the Repetition is not allowed, the number at Ten's place must not be the No. at Unit's place.

The Ten's place can be filled from the remaining allowed digits

$\square\quad \! \! \! \underbrace{\blacksquare}_\substack{\rm Except\\\rm U}}\quad\footnotesize{\boxed{\rm U}}\\\scriptsize{\boxed{\textsf{Where U can be 1/2/3}}}$

No. of ways of ways of filling the Ten's place = 5

Remaining Allowed Digits = 4

$\underline{\LARGE{\textsf{Step-3}}}$
Find the Number of ways of Filling the Hundred's Place.

As the Repetition is not allowed, the number at Hundred's place must neither be the No. at Ten's Place nor be the No. at Unit's Place.

$\quad \underbrace{\blacksquare}_\substack{\rm Except\\\rm U\: and\: T}}\! \! \! \quad\footnotesize{\boxed{\rm T}}\quad\footnotesize{\boxed{\rm U}}\\\scriptsize{\boxed{\textsf{Where U can be 1/2/3}}}$

No. of ways of ways of filling the Hundred's place = 4

$\underline{\LARGE{\textsf{Step-4}}}$
$\textsf{Find the No. of Numbers that can be formed}\\\sf\textsf{using the digits 1, 2, 3 4, 5, 6 when repetition of }\\\sf\textsf{the digits is allowed.}$

$\blacktriangle$No. of Numbers that can be formed is the product of No. of possible ways of filling each place in the 3-digit Number.

$\therefore \sf\textsf{No. of Numbers that can be formed from} \\ \sf \textsf{given digits \textbf{without Repetition}} = \mathsf{4\times 5\times 3} = \textbf{60}$



$\underline{\Huge{\textsf{Case-b}}}$
$\underline{\textit{Repetition is allowed.}}$

$\underline{\LARGE{\textsf{Step-1}}}$
$\textsf{Find the Number of ways of Filling}\\\textsf{the Unit's Place}$

As the Number to be formed must be odd number, the unit digit of the number must be an odd number from the allowed digits.

Odd No.s from the Allowed digits = 1, 3, 5

$\square\quad\square\quad \! \! \! \underbrace{\! \! \blacksquare\! \!}_{1/2/3}$

No. of ways of ways of filling the Unit's place = 3

$\underline{\LARGE{\textsf{Step-2}}}$
$\textsf{Find the Number of ways of Filling the}\\\textsf{Ten's Place.}$

As the Repetition is allowed, the number at Unit's place can also be used at Ten's Place.

The Ten's place can be filled from allowed digits.

$\square\quad \! \! \! \underbrace{\blacksquare}_\substack{\rm Any\: of \\\rm Allowed\: Digits}}\quad\footnotesize{\boxed{\rm U}}\\\qquad \scriptsize{\boxed{\textsf{Allowed Digits = 1/2/3/4/5/6}}}$

No. of ways of ways of filling the Ten's place = 6

$\underline{\LARGE{\textsf{Step-3}}}$
$\textsf{Find the Number of ways of Filling the}\\\textsf{Hundred's Place.}$

As the Repetition is not allowed, the number at Hundred's place can also be the No. at Ten's Place or be the No. at Unit's Place.

$\underbrace{\blacksquare}_\substack{\rm Any\: of \\\rm Allowed\: Digits}}\! \! \quad\footnotesize{\boxed{\rm T}}\qquad \footnotesize{\boxed{\rm U}}\\\qquad \scriptsize{\boxed{\textsf{Allowed Digits = 1/2/3/4/5/6}}}$

No. of ways of ways of filling the Hundred's place = 6

$\underline{\LARGE{\textsf{Step-4}}}$
$\textsf{Find the No. of Numbers that can be formed}\\\sf\textsf{using the digits 1, 2, 3 4, 5, 6 when repetition of }\\\sf\textsf{the digits is allowed.}$

$\blacktriangle$No. of Numbers that can be formed is the product of No. of possible ways of filling each place in the 3-digit Number.

$\therefore \textsf{No. of Numbers that can be formed from}\\\textsf{given digits \textbf{with Repetition}} = \mathsf{6\times 6\times 3} = \textbf{108}$

$\blacksquare\; \; \textsf{No. of Odd Numbers that can be formed}\\\textsf{from the digits 1, 2, 3, 4, 5, 6}\\\blacktriangleright \: \textbf{without Repetition = \underline{\Large{ \: 60 \: }}}\\\blacktriangleright \: \textbf{with Repetition\: \quad \: = \underline{\Large{ \: 108 \: }}}$