Question

An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without replacement. The number of ways I which this can be done is  

Solution

Correct option is

(36)(7!)

We have 0 + 1 + 2 + 3 … + 8 + 9 = 45. 

To obtain an eight digit number exactly divisible by 9, we must not use either (0, 9) or (1, 8) or (2, 7) or (3, 6) or (4, 5). [Sum of the remaining eight digits is 36 which is exactly divisible by 9.] 

When, we do not use (0, 9), then the number of required 8 digit numbers is 8!. 

When one of (1, 8) or (2, 7) or (3, 6) or (4, 5) is not used, the remaining digit can be arranged in 8! – 7! Ways as  0 cannot be at extreme left. 

Hence, there are 8! + 4(8! – 7!) = (36)(7!) numbers in the desired category.

SIMILAR QUESTIONS

Q1

Let n = 2009. The least positive integer k for which

  

for some positive integer r is  

Q2

If 0 < r < s ≤ n and nPr = nPs, then value of r + s is

Q3

, then value of is

Q4

 is maximum where m is   

 

Q5

The number of rational numbers lying in the interval (2008, 2009) all whose digits after the decimal point are non-zero and are in deceasing order is

Q6

The number of positive integral solutions of the equation  is

Q7

The exponent of 7 in 100C50 is

Q8

In the certain test there are n questions. In this test 2k students gave wrong answers to at least (n – k) questions, where k = 0, 1, 2, …, n. If the total number of wrong answer is 4095, then value of n is 

Q9

If n > 1 and n divides (n – 1)! + 1, then 

Q10

In a group of 8 girls, two girls are sisters. The number of ways in which the girls can sit so that two sisters are not sitting together is