In a certain test,$a_i$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, 3, \dots, k$. No student gave more than $k$ wrong answers. The total number of wrong answers given is:

In a survey,it was found that $21$ people liked product $A$,$26$ liked product $B$,and $29$ liked product $C$. If $14$ people liked products $A$ and $B$,$12$ people liked products $C$ and $A$,$14$ people liked products $B$ and $C$,and $8$ liked all three products,find how many people liked product $C$ only.

In a certain town,$25\%$ of families own a phone,$15\%$ own a car,and $65\%$ of families own neither a phone nor a car. If $2000$ families own both a car and a phone,consider the following statements:
$1$. $10\%$ of families own both a car and a phone.
$2$. $35\%$ of families own either a car or a phone.
$3$. $40,000$ families live in the town.
Which of the above statements are correct?

$A$ number is chosen at random from the set $\{1, 2, 3, \ldots, 2000\}$. Let $p$ be the probability that the chosen number is a multiple of $3$ or a multiple of $7$. Then the value of $500p$ is . . . . . .

In a group of students,$100$ students know Hindi,$50$ know English,and $25$ know both. Each student knows either Hindi or English. How many students are there in the group?

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.