Question

Four people sit round a circular table, and each person will roll a normal six sided die once. The probability that no two people sitting next to each other will roll the same number is $\frac{N}{1296}$.

Answer 1

Answer: 630

$n(S)=6\cdot 6\cdot 6\cdot 6=6^4=1296$
$n(A)$ : Case-1- Suppose A & C throws the same number.
A can throw die in 6 ways $\Rightarrow C$ can throw only in one way.
$B$ can throw only in 5 ways and
D can also throw in 5 ways
Therefore number of ways are $6\times 5\times 1\times 5=150$ ways.

Case-2 -Suppose A & C throws different number.
A can throw in 6 ways $\Rightarrow C$ can throw in 5 ways
Now $B$ can throw in 4 ways and (Both cannot throw the number which $A\&C$ has thrown)
D can throw in 4 ways
Hence number of ways $=6\times 4\times 5\times 4=480$
Hence $n(A)=150+480=630$
$\Rightarrow$ probability is $\frac{630}{6^4}\Rightarrow N=630$