Question

Ten books are arranged in a line on a bookshelf in how many ways can we select 4 books such that no two conseutivwe books from the shelf are chosen

Answer 1

Answer:

Let the books be numbered as 1,2,3,..........,10. Consider one such selection in which no two books are consecutive { 3,6,8,10}. Associate with this selection a binary sequence in which '0' appears in the position if the book is absent else '1'. The corresponding binary sequnce for the selection {3,6,8,10} is 0,0,1,0,0,1,0,1,0,1. Observe that no two 1's appear consecutively since now two books are chosen consecutively. So for every such selection we have such a sequnce with the property that no two 1's appear consecutively. Conversely for every such sequence there will be one such selection. For consider the sequence 1,0,0,1,0,1,0,0,1,0 gives a selection {1,4,6,9}.

So the problem boils down to find such sequnce of 4 1's and 6 0's in which no two 1's appears consecutively. For this consider

* 0 * 0 * 0 * 0 * 0 * 0 * where the '*' represent the vacant position in which 1 can be placed such that no two 1's appears consecutively. There are 7 such places out of which 4 places can be chosen in 7C4 ways = 35 required selection

Step-by-step explanation: