In how many ways can 5 different books be arranged on a shelf if(i) th

1.	In how many ways can 5 different books be arranged on a shelf if(i) there are no restrictions(ii) 2 books are always together(iii) 2 books are never together
Answer» (i) 5 books arranged in ⁵P₅ = 5! = 120 ways. (ii) 2 books are together. Let us consider two books as one unit. This unit with the other 3 books can be arranged in⁴P₄ = 4! = 24 ways. Also, two books can be arranged among themselves in ²P₂ = 2 ways. ∴ Required number of arrangements = 24 × 2 = 48 (iii) Say books are B₁ , B₂ , B₃ , B₄ , B₅ are to be arranged with B₁ , B₂ never together. B₃ , B₄ , B₅ can be arranged among themselves in ³P₃ = 3! = 6 ways. B₃ , B₄ , B₄ create 4 gaps in which B₁ , B₂ are arranged in ⁴P₂ = 4 × 3 = 12 ways. ∴ Required number of arrangements = 6 × 12 = 72

In how many ways can 5 different books be arranged on a shelf if(i) there are no restrictions(ii) 2 books are always together(iii) 2 books are never together

Answer»

(i) 5 books arranged in ⁵P₅ = 5! = 120 ways.

(ii) 2 books are together.

Let us consider two books as one unit. This unit with the other 3 books can be arranged in⁴P₄ = 4! = 24 ways.

Also, two books can be arranged among themselves in ²P₂ = 2 ways.

∴ Required number of arrangements = 24 × 2 = 48

(iii) Say books are B₁ , B₂ , B₃ , B₄ , B₅ are to be arranged with B₁ , B₂ never together.

B₃ , B₄ , B₅ can be arranged among themselves in ³P₃ = 3! = 6 ways.

B₃ , B₄ , B₄ create 4 gaps in which B₁ , B₂ are arranged in ⁴P₂ = 4 × 3 = 12 ways.

∴ Required number of arrangements = 6 × 12 = 72