Question

If $4$ digit numbers not less than $5000$ are randomly formed from the digits $0,1,3,5$ and $7$, what is the probability of forming a number divisible by $5$ when
(i) the digits are repeated?
(ii) the repetition of digits is not allowed?

Answer 1

We need numbers more than are equal to 5000 :

(a) if repetition is allowed then $\underset{–}{a}\underset{–}{b}\underset{–}{c}\underset{–}{d}$

Divisible by $5\Rightarrow d=0or5$

and abcd $\ge 5000\Rightarrow a=5$ OR $7$

and $b,c=0,1,3,5,$ or $7$

$\therefore$ Ans = $\left(2\right).\left(5\right).\left(5\right).\left(2\right)$

$=100$ Answer

(b) if repetition is not allowed then

case (1) $\Rightarrow a=5$ then $d=0$

$\therefore 5??0\Rightarrow {}^3{C}_2.2!=6$

Case (2) $\Rightarrow a=7$ then d = 0 or 5

$\therefore 7??0/5\Rightarrow 2{(}^3{C}_2.2!)=12$

$\therefore 6+12=18$