The total number of numbers, lying between 100 and 1000 that can be formed with the digits 1,2, 3, 4, 5, if the repetition of digits is not allowed and numbers are divisible by either 3 or 5, is__________.     

The given digits are 1,2,3,4,5

the number of three digit numbers which are divisible by 5: 

in three blanks, first fill unit place with 5, this can be done in only one way

Remaining two blanks can be filled with the remaining four digits in $4\times 3=12$ ways

therefore, the number of three digit numbers which are divisible by 5, formed by the given digits is 12. 

 

The number of three digit numbers which are divisible by 3: 

Select the combination of three numbers such that whose sum is divisible by 3

case (i) : 1,2,3 … arrange among them selves, this can be done in 6 ways 

case (ii): 1,3, 5 … arrange among them selves, this can be done in 6 ways ( some of these numbers repeated with three digit numbers having 5 in unit place, only 2 are here)

Case (iii): 2,3, 4 … arrange among them selves, this can be done in 6 ways

Case(iv) 3,4,5 … arrange among them selves, this can be done in 6 ways ( some of these numbers repeated with three digit numbers having 5 in unit place, only 2 are here)

Hence the number of three digit numbers which are divisible by 3  is 24

 

therefore, the required number is $24+12-4=32$