Question

How many numbers can be formed from the digits $1,2,3,4$ when the repetition is not allowed

• A. ${}^4{P}_4$
• B. ${}^4{P}_3$
• C. ${}^4{P}_1{+}^4{P}_2{+}^4{P}_3$
• D. ${}^4{P}_1{+}^4{P}_2{+}^4{P}_3{+}^4{P}_4$

Answer 1

The correct option is A ${}^4{P}_4$

I can choose the units place digit ; any of

the four ways 1,2,3,4 if repetition is not

allowed ie. $xyz\underset{–}{1},\underset{–}{x}\underset{–}{y}\underset{–}{z}\underset{–}{2},\underset{–}{x}\underset{–}{y}\underset{–}{2}\underset{–}{3},\underset{–}{x}\underset{–}{y}\underset{–}{2}\underset{–}{4}$

$\Rightarrow \underset{–}{}\underset{–}{}\underset{–}{}\underset{–}{4}$

For the ten's paced digit; I can choose any 3

of 4 ways 1,2,3,4 if repetition is not allowed

$\therefore 4\times 3\Rightarrow \underset{–}{}\underset{–}{}\underset{–}{3}\underset{–}{4}$

For the hundred's place digit I can choose any

2 of 4 ways 1,2,3,4 is arranged to avoid up etc then

$\Rightarrow \underset{–}{}\underset{–}{2}\underset{–}{3}\underset{–}{4}$

For the thousands digit place I can any choose

one digit $\underset{–}{1}\underset{–}{2}\underset{–}{3}\underset{–}{4}$ (if npo repetition allowed )

Therefore, total no. of numbers that can we

formed from digit 1,2,3 and 4 when repetition

is not allowed is $1\times 2\times 3\times 4\Rightarrow 24\Rightarrow 4!$

also, ${}^4{P}_4\Rightarrow \frac{4!}{0!}\Rightarrow 24$ Ans