Question

On New Year's day, every student of a class sends a card to every other student. The postman delivers $600$ cards. The number of students in a class are

1. 42
2. 34
3. 25
4. 52

Answer 1

Every student will deliver card to all students except the student himself.
This indicates that the number of students to whom a card is delivered by every student is one less than the total number of students.

Complete step by step answer:
Let the number of students in the class be x.
As each student delivers a card to every other student, the number of students to whom card is delivered are equal to (x – $1$).
Total number of cards delivered is calculated as the product of the number of students who delivered the card and number of students to whom the card has been delivered.
→ Total cards delivered = x × (x - $1$)
$\begin{aligned} & 600=x\times (x-1) \\\ & {{x}^{2}}-x=600 \\\ & {{x}^{2}}-x-600=0 \end{aligned}$
The equation obtained in variable x is a quadratic equation as the highest power of x is $2$. The middle-term splitting factor method can be applied to find the solutions of the above equation.
The equation is solved as:
$\begin{aligned} & {{x}^{2}}-x-600=0 \\\ & {{x}^{2}}-(25-24)x-600=0 \\\ & {{x}^{2}}-25x+24x-600=0 \\\ & x(x-25)+24(x-25)=0 \\\ & (x-25)(x+24)=0 \\\ & x=25 \end{aligned}$
Two factors of equation came out to be $(x-25)$ and $(x+24)$. When factor $(x+24)$is put equal to zero, it gives a value of x equal to $-24$ which is not appropriate as the number of students cannot be negative.
Factor $(x-25)$when put equal to zero gives value of x equal to $25$. This indicates that there are $25$students in a class.

So, the correct answer is “Option 3”.

Note: If the number of students is multiplied by themselves, the answer obtained would be incorrect as the student cannot deliver the card to himself.
The cards delivered do not have to be decided by order as each student is performing the same action.