Question

On the occasion of Deepawali festival each student of a class sends greeting cards to the others. If there are $20$ students in the class, then the total number of greeting cards exchanged by the students is

• A. ${}^{20}C_2$
• B. $2\times {}^{20}C_2$
• C. $2\times {}^{20}P_2$
• D. None of these

Answer 1

The correct option is B

$2\times {}^{20}C_2$

Explanation for the correct option

Determining the total number of cards exchanged

Given, there are $20$ students in the class.

Since the total number of students in the class is $20$, so each students will give greetings cards to other $19$ students.

So, according to the fundamental principle of counting, the total number of greetings cards exchanged will be:

$$\begin{gathered} N=20\times 19 \\ =2\times \frac{20\times 19}{2} \\ =2\times \frac{20\left(20-1\right)}{2!} \\ =2\times \frac{20.19.18.17....1}{2!\left(18.17.16.15.....1)\right)} \\ =2\times \frac{20!}{2!\times 18!} \\ =2\times {}^{20}C_2\left[\because {}^{n}C_2=\frac{n!}{r!(n-r)!}\right] \end{gathered}$$

Hence, the correct answer is Option (B).