Solveeit Logo
Login

Question

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

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

Explanation

Solution

To solve the above question, we will form different cases in which we will take n=1,2,3,....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 x1,x2,x3.....xn{{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 x1{{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=2 .
Case II: - Let us assume that we have to distribute x1{{x}_{1}} and x2{{x}_{2}} into the two distinct boxes. Here, both x1{{x}_{1}} and x2{{x}_{2}} may go to first box and x2{{x}_{2}} in second one or x2{{x}_{2}} in first box and x1{{x}_{1}} in second one. So, the number of ways of doing this are =4=4.
So, here we can see that when we have taken 1 object, the number of ways we get are =2(=21)=2\left( ={{2}^{1}} \right) . When we take 2 objects the number of ways, we will get =4(=22)=4\left( ={{2}^{2}} \right) . Thus, when we will take 3 the number of ways we will be getting are =23={{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 =2n={{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.