Among the 4-digit numbers that can be formed using the digits $1,2,3,4,5,6$ without repeating any digit, the number of numbers which are divisible by 6 is

Since number divisible by 6 i.e. divisible by 2 and 3 . For divisible by 2 units place of number contains $2,4,6$. For divisible by 3 , the sum of digits is divisible by 3 . Case I: If units place is 2 . Then rest 3 digits may be $(1,4,5),(1,3,6),(3,4,6)$ $\therefore$ Number of number with units place $2=3.3!=18$ Case II : If units place is 4 Then rest 3 digits may be $(1,2,5),(2,3,6),(3,5,6)$ $\therefore$ Number of number with units place $4=3.3!=18$ Case III : If units place is 6 Then rest 3 digits may be $(1,2,3),(2,3,4),(3,4,5)$ and $(1,3,5)=4.3!=24$ Total number divisible by $6=18+18+24=60$.