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

$n(S) = 6 · 6 · 6 · 6 = 6^4 = 1296$
$n ( A )$ : 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 $\frac{630}{6^4} \Rightarrow N =630$