Question

Find the total number of ways in which n distinct objects can be put into two different boxes.

Answer 1

Hint: To solve the above question, we will form different cases in which we will take $n=1,2,3,....$ where n is the number of distinct objects. After taking a number of distinct objects as 1, 2 and 3, we will try to predict the pattern or function it is following in these cases and with help of them, we will determine the value when n distinct objects are taken.

Complete step-by-step answer:
In the above question, it is given that both the objects and the boxes are distinct in nature. Also there is no restriction on no condition on how we can put n distinct objects into two different boxes. So, to solve the question, let us check how many choices does each box have:
Let the objects be named \[{{x}_{1}},{{x}_{2}},{{x}_{3}}.....{{x}_{n}}\] . Now, we are going to consider some cases here for each of the objects.
Case I: - Let us assume that we have to distribute x, into the two distinct boxes. There are two methods of doing this, either we can put ${{x}_{1}}$ in the first box or we can put it in the second box. So, the number of ways of doing this are $=2$ .
Case II: - Let us assume that we have to distribute ${{x}_{1}}$ and ${{x}_{2}}$ into the two distinct boxes. Here, both ${{x}_{1}}$ and ${{x}_{2}}$ may go to first box and ${{x}_{2}}$ in second one or ${{x}_{2}}$ in first box and ${{x}_{1}}$ in second one. So, the number of ways of doing this are $=4$.
So, here we can see that when we have taken 1 object, the number of ways we get are $=2\left( ={{2}^{1}} \right)$ . When we take 2 objects the number of ways, we will get $=4\left( ={{2}^{2}} \right)$ . Thus, when we will take 3 the number of ways we will be getting are $={{2}^{3}}$ .
Similarly, when we take n objects and put them into the 2 distinct boxes, the number of ways we will be getting is equal to $={{2}^{n}}$ .

Note: Here, we cannot use the partition method. Although, it is used when there is no restriction but it is used only when we are considering objects as identical.