Two numbers $a$ and $b$ are chosen at random from the set of first $30$ natural numbers. The probability that ${a^2} - {b^2}$ is divisible by $3$ is

$n$ cadets have to stand in a row. If all possible permutations are equally likely, then the probability that two particular cadets stand side by side, is

A dice marked with digit $\{1, 2, 2, 3, 3, 3\} ,$ thrown three times, then the probability of getting sum of number on face of dice is six, is equal to :-

If $10$ different balls are to be placed in $4$ distinct boxes at random, then the probability that two of these boxes contain exactly $2$ and $3$ balls is 

There are $10$ engineering colleges and five students $A, B, C, D, E$ . Each of these students got offer from all of these $10$ engineering colleges. They randomly choose college independently of each other. Tne probability that all get admission in different colleges can be expressed as $\frac {a}{b}$ where $a$ and $b$ are co-prime numbers then the value of $a + b$ is

Let $S=\{1,2,3,4,5,6\} .$ Then the probability that a randomly chosen onto function $\mathrm{g}$ from $\mathrm{S}$ to $\mathrm{S}$ satisfies $g(3)=2 g(1)$ is :