An eight-digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is :

S=0,1,2,3,4,5,6,7,8,9→10digits∑⁹ᵢ₀i=45→divisible by 9∴For 8 digit number we need to remove two digits from SAfter removing∑→divisible by 9∴We can only remove the pairs(0,9),(1,8),(2,7),(3,6),(4,5)Since 0+9=9, 45−9=36→divisible by 91+8=9, 45−9=36→divisible by 94+5=93+6=9∴If(0,9)are removed then no. of 8 digits nos possible=8!if(1,8)are removed the no. of 8 digit nos=8!(subtracting the number of cases where '0' is at the left most place)Similarly, when we remove(2,7),(3,6)and(4,5)we get 8! in each case.∴Total 8 digit nos=8!+4(8!)=5⋅8!=5⋅8⋅7!=40⋅7!8⋅7!=36(7!)