From a month of $31$ days, $3$ different dates are selected at random. If the probability that these dates are in an increasing $A$.$P$. is equal to $\frac{a}{b}$, where $a, b \in N$ and $\text{gcd}(a, b) = 1$, then $a + b$ is equal to ————

$A$ bag contains $a$ white and $b$ black balls. Two players $A$ and $B$ alternately draw a ball from the bag,replacing the ball each time after the draw,until one of them draws a white ball and wins the game. $A$ begins the game. If the probability of $A$ winning the game is three times that of $B$,then the ratio $a : b$ is

$A$ bag contains $6$ white and $4$ black balls. Two balls are drawn at random. The probability that they are of the same colour is

Two cards are drawn at random from a standard pack of $52$ cards. The probability that the draw consists of exactly one face card and exactly one ace card is

$A$ bag contains $6$ white and $4$ black balls. Two balls are drawn at random. The probability that they are of the same colour is . . . . . .

$A$ committee of $5$ members is to be formed from $6$ men and $4$ women. What is the probability that the committee contains at least one woman?