4 votes 4 votes Consider a bookshelf with $15$ books placed in sequential manner. In how many ways one can choose a set of $5$ books from this shelf so that no $2$ books in this set should be adjacently positioned at the time of picking $?$ Combinatory engineering-mathematics + – saxena0612 asked Jan 17, 2018 saxena0612 415 views answer comment Share Follow See all 6 Comments See all 6 6 Comments reply Show 3 previous comments Higgs commented Jan 17, 2018 reply Follow Share //Just had a thought... This problem can be reframed as: Map 5 books to 15 positions such that no two books are adjacent. which can be further reframed as: Map 5 objects to 15 positions such that no two objects are adjacent. which can be further reframed as: No. of bit strings of length 15 such that exactly 5 bits are 1 and bits that are 1 can't be adjacent. Eg: Bit string 101010101000000 can be interpreted as: book1 is placed in pos. 1 (assuming we count positions from the lhs of the bit string.) book2 is placed in pos. 3 and so on.... This problem can be further reframed as: 15 bits = 5 1's + 10 0's Pos. of 10 0's is fixed: i.e. _0_0_0_0_0_0_0_0_0_0_ In how many ways, 5 1's can be mapped to 11 positions. (blanks seen above.) So, our problem further reduces to: In how many ways 5 elements be chosen from 11 elements? //elements = positions. This can be done in: $\binom{11}{5}$ ways. 5 votes 5 votes joshi_nitish commented Jan 17, 2018 reply Follow Share @Higgs perfect !! 2 votes 2 votes minal commented Jan 17, 2018 reply Follow Share https://gateoverflow.in/126428/rosen 5 votes 5 votes Please log in or register to add a comment.