Sum of numbers

Question

Hello I was readig through these questions: permutation-88357.html?hilit=digit%20using manhattan-arithimetic-chapter-1-need-help-88869.html?hilit=digit%20using#p670812 sum-of-all-3-digit-nos-with-88864.html?hilit=digit%20using m04-70602.html?hilit=digit%20using testing-number-properties-91007.html?hilit=digit%20using can-someone-help-94836.html?hilit=repetition and I was wondering, how could you solve a question, with 0 in the digits? For examlpe: what is the sum of all 4 digit numbers formed from numbers: 0,1,2,3 Or seccond: formend from numbers: 0,1,2,3,4 Thanks

passionatevibes · Answer

The total number of distinct nos that can be formed using 0,1,2,3 is the same as the number of ways in which you can arrange these numbers, which is 4! (24). This includes nos that have 0 in the thousands place.

So nos are like these:
0XXX
1XXX
2XXX
3XXX
Observe that each digit occurs 24/4=6 times in the thousands place. This is true for the hundreds, tens and unit's place.

So, each column adds up to 6(0+1+2+3)=36.

Hence, sum is 36(1000+100+10+1)=39996

This also includes numbers that start with the digit 0. So, we have to subtract the sum of such numbers that start with zero from the sum 39996.

Now, there are 3!=6 numbers that start with 0 in the thousands place. And each such number will have 1,2,3 repeated 2 times. So the sum of each column is 2(1+2+3)=12.
The sum of all such numbers is 12(100+10+1)=1332

Subtract 1332 from the actual sum and that gives you the answer: 39996-1332=38664

craky · Answer

Very nice! Can you do that also for the case, when repetition is allowed?

passionatevibes · Answer

When repetition is allowed:
Here we find sum of all 4 digit (including starting with 0) and subtract where total includes number starting with 0 or starting with 00 or with 000

When repetition is allowed, there can be 4^4 numbers possible, i.e, 256
Each digit is repeated 256/4=64 times in these numbers.
Thus, the sum of each column is 64(0+1+2+3)=384.

Hence, sum is 384(1000+100+10+1)=426624

Now,calculate the sum of all numbers formed by 0 as the first digit
(this covers three digit, 2 digit and single digit nos )

We can form 1* 4*4*4 = 64 such nos where repetition is allowed

In this 64 nos the 1st digit can be one or 0,1,2,3 or each digit has 4 possibilities

So in each column (when we do the sum of these numbers) 0 repeats 64/4 times = 16 times 1 repeats 16 times, 2 repeats 16 times and 3 repeats 16 times
Total of each column is (0+1+2+3)*16 = 96

Hence, sum is 96(100+10+1)=10656

Required sum is 426624-10656=415968

Sum of numbers

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