Sum these numbers

Number Theory Level 3

The digits 1,2,3 and 4 are used to generate 256 different 4 digit numbers.Find the sum of all these numbers.


The answer is 711040.

Naitik Sanghavi by naitik sanghavi , 20, India

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3 solutions

Vignesh Rao
Dec 25, 2015

Using the formula:- x y × z × ( 111.. . y times ) \frac{x}{y} \times z \times (111..._{y \text{ times}})

Where,

x x = No of numbers formed

y y = No. of digits given

z z = Sum of given digits

⇒ 256 4 ∗ ( 1 + 2 + 3 + 4 ) ∗ 1111 = 711040 \Rightarrow \frac{256}{4}*(1+2+3+4) * 1111 =\boxed{ 711040}

3 Helpful 0 Interesting 0 Brilliant 0 Confused

From where did you find that formula?

Shreyansh Mukhopadhyay - 3 years, 2 months ago

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Can be found using some combinatorics. Not a general formula as such.

Vignesh Rao - 3 years, 2 months ago 
1
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sum1 = 0
for a in xrange(1,5):
    for b in xrange(1,5):
        for c in xrange(1,5):
            for d in range(1,5):
                sum1 += ( 1000*a + 100*b + 10*c + d )

print sum1

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Nguyen Thanh Long
Jul 22, 2015

It is easy to get the result: 64 × ( 1000 + 100 + 10 + 1 ) × ( 1 + 2 + 3 + 4 ) = 711040 64 \times (1000+100+10+1) \times (1+2+3+4) = \boxed{711040}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

How to get that? Can you explain it?

Anik Mandal - 5 years, 10 months ago

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