If a coin is tossed seven times, then the probability of getting exactly three heads such that no two heads occur consecutively is

Cards are drawn one by one at random from a well-shuffled full pack of $52$ cards until two aces are obtained for the first time. If $N$ is the number of cards required to be drawn,then $P(N = n)$,where $2 \le n \le 50$,is

If the coefficients $a$ and $b$ of a quadratic expression $x^2+ax+b$ are chosen from the sets $A=\{3, 4, 5\}$ and $B=\{1, 2, 3, 4\}$ respectively, then the probability that the equation $x^2+ax+b=0$ has real roots is

$A$ box contains $50$ tickets numbered $1, 2, 3, \dots, 50$. Five tickets are drawn at random and arranged in ascending order $(x_1 < x_2 < x_3 < x_4 < x_5)$. What is the probability that $x_3 = 30$?

Let $A$ and $B$ be two finite sets having $m$ and $n$ elements respectively such that $m \le n.$ $A$ mapping is selected at random from the set of all mappings from $A$ to $B$. The probability that the mapping selected is an injection 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