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Combining Integers to generate a specific value










































Sum of Integers

Generate Integers from a Linear Combination of Integers.

     • Write each number listed below as a sum of integers in three different ways:

         1. -5

         2. +15

         3. 0

Comments for Combining Integers to generate a specific value

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Sep 06, 2012
Sum of Integers
by: Staff


Answer:
Part I


     • Write each number listed below as a sum of integers in three different ways:

         1. -5

             -20+5+5+5 = -5

             100-150+155-110 = -5

             -20+5+5+5+100-150+155-105 = -5




         2. +15

             1+2+3+4+5 = +15

             -1+2-3+4-5+6-7+8-9+10-11+12-13+14-15+16-17+18-19+20-21+22-23+24-25+26-27+28-29+30 = +15

             11+1+1+1+1 = +15




         3. 0

             0+0+0 = 0

             21-21+44-44 = 0

             -5+2+3 = 0


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Sep 06, 2012
Sum of Integers
by: Staff


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Part II

The integer combinations shown above should answer your immediate question.

However, there are some very interesting things you can think about to get a specific value by combining different integers.

For example, suppose you could only choose integers from the following set of eleven integers: {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}.

Now suppose you can choose as many integers as you want, but you can only use each integer once.

How many different combinations of integers from the choices available will yield a value of 5?

How many possible combinations are there?

If you choose only one integer, you have 11 possible choices.

If you choose two integers, you have 11*10 possible combinations.

If you choose three integers, you have 11*10*9 possible combinations.

If you choose four integers, you have 11*10*9*8 possible combinations.

If you choose five integers, you have 11*10*9*8*7 possible combinations.

If you choose six integers, you have 11*10*9*8*7*6 possible combinations.

If you choose seven integers, you have 11*10*9*8*7*6*5 possible combinations.

If you choose eight integers, you have 11*10*9*8*7*6*5*4 possible combinations.

If you choose nine integers, you have 11*10*9*8*7*6*5*4*3 possible combinations.


If you choose 10 integers, you have 11*10*9*8*7*6*5*4*3*2 possible combinations.

If you choose all 11 integers, you have 11*10*9*8*7*6*5*4*3*2*1 possible combinations.

Adding all these possible combinations together will produce a very large number of possibilities.

However, since you will adding the integers you have chosen, the order of the integers you choose will not matter. Adding the same integers in any order will produce the same result.

The number of possible combinations where order does not matter will be considerably less than the possibilities outlined above.

Since the order of the integers chosen does not matter.

n = number of elements in the set = 11 (eleven integer choices)

Number of possible subsets = 2ⁿ = 2^11 = 2048 possible integer combinations

How many of these integer combinations will add up to the number 5?

That will be a good project for someone interested in writing a computer program to solve it.



Thanks for writing.

Staff
www.solving-math-problems.com



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