The number of 6 -digit numbers, formed using distinct digits from the set 1, 2, 3, 4, 5, 6, 7, 8 , which are divisible by 99 , is

Step-by-step solution

A number is divisible by 99 if it is divisible by both 9 and 11 . For divisibility by 9 , the sum of the digits must be a multiple of 9 . The sum of all 8 digits in the given set is 36 . The sum of any 6 distinct digits from the set ranges from 1+2+3+4+5+6 = 21 to 3+4+5+6+7+8 = 33 . The only multiple of 9 in this range is 27 . Let the 6 -digit number be a₁ a₂ a₃ a₄ a₅ a₆ . Let S_ odd = a₁ + a₃ + a₅ and S_ even = a₂ + a₄ + a₆ . We have S_ odd + S_ even = 27 . For divisibility by 11 , S_ odd - S_ even must be a multi