Thursday, October 16, 2014

Find how many distinct numbers greater than 5000 and divisible by 3 can be formed from 3,4,5,6,0, each digit being used at MOST once in any number

We need to find the number of numbers we can form greater
than 5000 and divisible by 3.


Now we see that 3+4+5+6 = 18
which is divisible by 3 so any number which has all the digits will be divisible by
3.


Therefore we can have any 5 digit number. The number of
these possible is 4*4*3*2*1 = 72


Now to find the number of
4 digit numbers that satisfy this condition. We see that the only sets which are
divisible by 3 are 4,5,6,3 and 5,4,3,0 and 5,4,6,0


The
first digit can only be 5 or 6.


Now the numbers starting
with 5 and divisible by 3 are: 5604, 5640 , 5460 , 5406, 5064, 5064, 5304, 5340, 5034,
5043, 5403, 5430, 5643, 5634, 5436, 5463, 5346, 5364. We have 18
numbers


The numbers starting with 6 and divisible by 3 are:
6540, 6504, 6450, 6405, 6045, 6054, 6345, 6354, 6435, 6453, 6534, 6543. we have 12
numbers.


Therefore the total number of numbers is 18+ 12
+72 = 102


The required result is
102.

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