Let b 1 b 2 b 3 b 4 be a 4 -element permutation with b i ∈ 1 , 2 , 3 , … … … , 100 for 1 ≤ i ≤ 4 and b i ≠ b j for i ≠ j , such that either b 1 ,   b 2 ,   b 3 are consecutive integers or b 2 , b 3 , b 4 are consecutive integers. Then the number of such permutations b 1 b 2 b 3 b 4 is equal to ______.

Step-by-step solution

b i ∈ 1 , 2 , 3 … … … 100 Let A = set when b 1   b 2   b 3 are consecutive n A = 97 + 97 + … … + 97   added   98   times = 97 × 98 = 9506 Similarly when b 2   b 3   b 4 are consecutive n B = 97 × 98 = 9506 Now when b 1   b 2   b 3   b 4 are consecutive then n A ∩ B = 97 n A U B = n A + n B - n A ∩ B Number of required permutation = 9506 + 9506 - 97 = 18915