From a group of $10$ men and $5$ women,a four-member committee is to be formed such that it must contain at least one woman. The probability that the committee has more women than men is:

Three numbers are chosen at random without replacement from $\{1, 2, \ldots, 10\}$. The probability that the minimum of the chosen numbers is $3$ or their maximum is $7$ is:

$A$ bag contains $13$ red,$14$ green,and $15$ black balls. The probability of getting exactly $2$ black balls when pulling out $4$ balls is $P_1$. Now,the number of balls of each color is doubled,and $8$ balls are pulled out. The probability of getting exactly $4$ black balls is $P_2$. Then:

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$,where $p$ and $q$ are co-prime,then $q-p$ is equal to

Two numbers are selected at random from $40$ consecutive natural numbers. What is the probability that the sum of the numbers is odd?

The probability that a randomly chosen $5-digit$ number is made from exactly two digits is