Question

On a new year day every student of a class sends a card to every other student. The postman delivers 600 cards. The numbers of students in the class are
A. 42
B. 34
C. 25
D. None of these

Answer 1

Hint: To solve this question first let us assume the number of students to be x. then as given in the question every student of a class sends a card to every other student. So each student sends a card to $n-1$ students. By equating the obtained equation to 600 and simplifying the obtained equation we get the desired answer.

Complete step by step answer:
We have been given that on a New Year day every student of a class sends a card to every other student and the postman delivers 600 cards.
We have to find the number of students in the class.
Let the number of students in the class is n.
Now, as given in the question that every student of a class sends a card to every other student it means each student sends a card to $n-1$ students.
The postman delivers 600 cards. Then we will get
$\Rightarrow n\left( n-1 \right)=600$
Simplifying the above equation we will get
$\begin{align}
  & \Rightarrow {{n}^{2}}-n=600 \\
 & \Rightarrow {{n}^{2}}-n-600=0 \\
\end{align}$
The above obtained equation is a quadratic equation. So we will solve the equation by splitting the middle term. Then we will get
$\begin{align}
  & \Rightarrow {{n}^{2}}-\left( 25n-24n \right)-600=0 \\
 & \Rightarrow {{n}^{2}}-25n+24n-600=0 \\
\end{align}$
Now, taking the common terms out we will get
$\Rightarrow n\left( n-25 \right)+24\left( n-25 \right)=0$
Now, again taking common terms out we will get
$\Rightarrow \left( n-25 \right)\left( n+24 \right)=0$
Now, equating each factor to zero we will get
$\Rightarrow \left( n-25 \right)=0$ and $\left( n+24 \right)=0$
Simplifying both we will get
$\Rightarrow n=25$ and $n=-24$
As n is the number of students so it can’t take negative value. So we ignore the negative value. The number of students in the class is 25.
So, option C is correct.

Note: Students can give the correct answer as 24 by ignoring the sign, which is incorrect. Students can solve the obtained quadratic equation by using the quadratic formula which is given as $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . By comparing the equation with the general equation students easily get the values of a, b and c.