If $A = \{1, 2, 3, 4, 5, 6\}$,then the number of subsets of $A$ which contain at least two elements is

$A$ survey shows that $63\%$ of the Americans like cheese whereas $76\%$ like apples. If $x\%$ of the Americans like both cheese and apples,then

Leela and Madan pooled their music $CDs$ and sold them. They received as many rupees for each $CD$ as the total number of $CDs$ they sold. They shared the money as follows: Leela first takes $10$ rupees,then Madan takes $10$ rupees and they continue taking $10$ rupees alternately until Madan is left with less than $10$ rupees. Find the amount that is left for Madan at the end,with justification.

The number of $3$-digit numbers that are divisible by either $2$ or $3$ but not divisible by $7$ is $.........$.

An integer is chosen at random from the integers $\{1, 2, 3, \ldots, 50\}$. The probability that the chosen integer is a multiple of at least one of $4, 6,$ and $7$ is

Let $A = \{a_1, a_2, a_3, \dots, a_n\}$ be a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are formed independently. The number of ways in which these subsets can be formed such that $(P - Q)$ contains exactly $2$ elements is: