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 A Quicky III (Posted on 2014-04-09)
If each of the a,b,c,d is a decimal digit - for how many ordered quadruplets (a,b,c,d) is the sum S= a+b+6c+d divisible by 3 ?

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 How many are divisible by 1? | Comment 3 of 9 |
Well, let me solve the easier problem of just counting how many ordered quadruplets there are.

If all the digits are the same, then there are 10 quadruplets.

If there are only two different digits, then C(10,2)*3 = 135
The 3 multiplier is because the digits (1,2) can be either (1,1,1,2) or (1,1,2,2) or (1,2,2,2).

If there are three different digits, then C(10,3)*3 = 360
The 3 multiplier is because the digits (1,2,3) can be either (1,1,2,3) or (1,2,2,3) or (1,2,3,3).

If there are four different digits, then C(10,4) = 210.

Total quadruplets = 10 + 135 + 360 + 210 = 715

 Posted by Steve Herman on 2014-04-09 12:14:36

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