A seven-digit number without repetition and divisible by 9 is to be formed by using seven digits out of 1 , 2, 3, 4, 5, 6, 7, 8, 9. The number of ways in which this can be done is

Sum of seven digits is a multiple of 9. Now, sum of numbers 1, 2,3, 4, 5, 6,7 ,8, 9 is 45. So, we have to remove two numbers whose sum is 9. So, pairs of removed numbers are (1, 8), (2,7), (3,6), (4, 5).With each pair left number of 7-digit number is 7! . So with all 4 pairs, total numbers are 4$\times$7!.