Question

A palindrome is chosen randomly between 1000 and 10000 . If the probability that it is divisible by 7 is $\frac{ p }{10}$; the probability that it is divisible by 11 is $\frac{ q }{10}$ and probability that it is divisible by 13 is $\frac{ r }{10}$, find the value of $( p + q + r )$.

Answer 1

Palindrome is $x y y x$ where $x \in\{1,2, \ldots .8,9\}$ and $y \in\{0,9\}$
Now, $x$ y y $x=1000 x+100 y+10 y+x=1001 x+110 y$
Now, 1001 is divisble by 7 , hence $x$ can be anything from 1 to 9 .
For $110 y$ to be divisible by $7, y$ must be either 0 or 7 .
Hence, $n ( S )=10 ; n ( A )=2$
Probability $=\frac{2}{10}=\frac{ p }{10} \Rightarrow p =2$
Probability that it is divisible by 11 is $1=\frac{q}{10} \Rightarrow q=10$
Probability that it is divisible by 13 is $\frac{1}{10}=\frac{r}{10} \Rightarrow r=1 (y=0, y$ cannot be 13)
Hence $p + q + r =2+10+1=13$.