The number ways in which n distinct objects can be put into two distinct boxes so that no box remains empty, is

(A) 2^(n)-1

(B) n^2 -1

(C) 2^n -2

(D) n^2 -2

(E) n!

1st object can b place in 2 ways (either of the boxes)

Similarly, 2nd object can b place in 2 ways

Similarly, 3rd object can b place in 2 ways

Similarly, 4th object can b place in 2 ways

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.

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.Similarly, nth object can b place in 2 ways

Because all placement of 2nd object is independent of placement of 1st object, the total no of ways in n objects can be place in two boxes = 2 x 2 x 2 x 2.............2(nth)

= \(2^n\)

But this total no ways also include the possibility - 2 cases

1) that all the objects are placed in only box1

2) that all the objects are placed in only box2

Because we want all the boxed to be filled we need to reduce the total no of ways by 2. So

\(2^n\) - 2

Hope this helps many

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