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

Answer: 13

Palindrome is $xyyx$ where $x\in \{1,2,\dots .8,9\}$ and $y\in \{0,9\}$
Now, $x$ y y $x=1000x+100y+10y+x=1001x+110y$
Now, 1001 is divisble by 7 , hence $x$ can be anything from 1 to 9 .
For $110y$ 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 $p+q+r=2+10+1=13$.