# Four digit number permutations

#### chaoseverlasting

1. The problem statement, all variables and given/known data

Find the sum of all the four digit numbers that can be formed with the digits 0,1,2,3

2. Relevant equations

3. The attempt at a solution

The total number of numbers possible is 3*4*4*4=192.

Since the lowest number we can form is 1000, and the highest is 3333, the sum of all the digits should be summation (1-3333) - summation(1-1000).

The summation is given by $$\frac{n(n+1)}{2}$$ or in this case,
$$\frac{3333*3334-1000*1001}{2}$$ which gives 5055611.... which is nt the given answer. What am I doing wrong?

#### 3trQN

Wrt(1) Is that Permutation or combination you want? Because 1111 and 0123 are both 4digit numbers formed with the digits. I assume its a permutation by the title of your post though.

You know that the permutations is simply 4! = 24 when n=r ? and that sum of each is constant?

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#### d_leet

1. The problem statement, all variables and given/known data

Find the sum of all the four digit numbers that can be formed with the digits 0,1,2,3

2. Relevant equations

3. The attempt at a solution

The total number of numbers possible is 3*4*4*4=192.

Since the lowest number we can form is 1000, and the highest is 3333, the sum of all the digits should be summation (1-3333) - summation(1-1000).

The summation is given by $$\frac{n(n+1)}{2}$$ or in this case,
$$\frac{3333*3334-1000*1001}{2}$$ which gives 5055611.... which is nt the given answer. What am I doing wrong?
The number 1999, as an example, is included in your sum, but it is not one of the numbers that can be formed with only the digits 0,1,2,3.

#### chaoseverlasting

Oh yeah... youre right... I completely overlooked that....

The sum would be 1000-1333 + 2000-2333+ 3000-3333... is there some simpler way to calculate it besides the summation formula?

#### drpizza

Simpler? 6 times, the one's digit is a 3. 6 times, the one's digit is a 2. 6 times, the ones digit is a 1. 6 times, the 1's digit is a zero.

6 times, the ten's digit is a 3. 6 times, the ten's digit is a 2...

Think about it from there.

#### drpizza

maybe it'll help if I pointed out that 34 + 43 has the same sum as 33 + 44.

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