How many positive integers between 1000 and 9999 inclusive a

How many positive integers between 1000 and 9999 inclusive:

a.) have distinct digits?

b.) are not divisible by 3?

c.) are divisible by either 5 or 7?

d.) are not divisible by either 5 or 7?

e.) are divisble by 5 but not by 7?

f.) are divisible by 5 and 7?

we want distinct 4 digits numbers, so we are having digits 0-9(10 digits)

we want 4 digit number, 0 can\'t be at first digit place, so we have 1-9 digits i.e = 9 digits

for second digit out 10, we use 1 digit in first digit place, so we have 9 choices for second digit

for third digit we have 8 chioces

for 4th digit we have 7 chioces,

So, totally 9*9*8*7 combination digits

i.e = 4536

intergers between 1000 and 9999 are 9000

in that divisible by 3 are 9000/3

if we do complement wi well get not divisible by 3

i.e = (9000-(9000/3))

=6000

divisible by 5 are 9000/5=1800

divisible by 7 are 9000/7=1285.81=1286

divisible by both are 9000/35 = 257

but we want divisible by 5 or 7, so answer will be 1800+1286-257

= 2829

this is simple complement previous question (c)

i.e = 9000-2829

=6171

e) this is divisible 5 but not divisible by 35

divisible by 5 = 9000/5 =1800

divisible by 35 = 257

this is divisible 5 but not divisible by 7 is = 1800-257

=1543

these are only numbers divisible by 35

i.e= 9000/35 = 257