Question

A person has to send the marriage invitations to his 7 friends. If he has only 3 servants to carry them, in how many ways he can send the invitation?
A) ${3^7}$
B) ${7^3}$
C) $21$
D) None of these

Answer 1

Hint: In this type of question we use the sum and the product rule of counting. The product rule of counting is that when one event occurs in m ways and the other event occurs in n ways then both the event occurs in $m \times n$ ways. The number of ways of occurrence of one event or the other event is \[m + n\] ways this is the sum rule of counting. The event here is sending the invitations.

Complete step-by-step answer:
We have 3 servants to send the invitations to any one of the friends he can send any of the servants. So the total number of ways in which the person sends the invitation to any one of the friends is 3. Similarly for all the remaining friends, he can send the invitation in 3 ways. So three events have to done 7 times so applying the product rule of counting we can say that the number of ways in which the person can send the invitations to his seven friends is
$N = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
Using the rule of exponents, we can conclude that:
$N = {3^7}$

Therefore, the correct option is A.

Note: In the above problem, we have used the rule of exponents, which says that when any number $x$ is multiplied $a$ times then it can be expressed as ${x^a}$. That is,
${x^a} = x \times x \times x \cdot \cdot \cdot {\text{upto }}a{\text{ terms}}$
Remember that if the $x$ is added $a$ times then it can be given as:
$x + x + x + \cdot \cdot \cdot {\text{upto }}a{\text{ terms }} = ax$