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Solution


Q.

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ______ .          [2023]

Ans.

(21)

We have to make 4-digit numbers using the digits 1,2,3, and 5.

The unit digit of the 4-digit number will be 5.

x y z 5

Now, the sum (x+y+z) should be of the form (3λ+1).

Therefore, the possible cases are:

(x,y,z)=(1,1,5),(1,1,2),(2,2,3),(2,3,5),(3,3,1),(5,5,3)

So, total arrangements are:

For (1,1,5)→3!2!=3  ;  For (1,1,2)→3!2!=3

For (2,2,3)→3!2!=3  ;  For (2,3,5)→3!=6

For (3,3,1)→3!2!=3  ;  For (5,5,3)→3!2!=3

So, total number of arrangements =3+3+3+6+3+3=21