discrete math

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:
(a) are divisible by 5 and by 7.


(b) have distinct digits.


(c) are not divisible by either 5 or 7.

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  1. (a) div. by 5 and by 7 => div. by 35
    We can find out that
    35*286=10010 and 35*29=1015
    Therefore the number divisible by 35, n35
    = 286-29=257

    (b) distinct digits
    First digit has 9 choices (1-9)
    second and subsequent digits 9,8,7 choices each
    Numbers with distinct digits
    = 9*9*8*7
    = 4536

    (c) not divisible by either 5 or 7
    Divisible by 5, n5= (10000-1000)/5=1800
    Divisisble by 7, n7 = (10003-1001)/7=1286
    Divisible by 5 or 7 or both
    =n5+n7-n35
    =1800+1286-257
    =2829

    Numbers NOT divisible by either 5 or 7
    =(10000-1000)-2829
    =6171

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