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General formula to calculate number ways

 
 
muna27
 
Reply Wed 2 Jul, 2014 01:37 am
Q: How many ways some integers (with repetition) can be added to get a particular sum. For example if my total sum is 4 and by using the integers 1,2,3 and 4 how many ways I can add these numbers to get 4. Order is not important. To get sum 4 the possible ways are give below.
1: 4 =4;
2: 3+1=4
3: 1+3=4;
4: 2+2=4
5: 1+1+2=4
6: 1+2+1=4
7: 2+1+1=4
8: 1+1+1+1+1=4;

So the total number ways is 8. For bigger sum like 100 there is is huge number ways by using the integers 1 to 100. Is it possible to derive a general formula ?
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Type: Question • Score: 2 • Views: 1,103 • Replies: 3
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engineer
 
  1  
Reply Wed 2 Jul, 2014 05:25 am
@muna27,
It's a binomial expansion.

2 : 1
3: 1 + 2 + 1
4: 1 + 3 + 3 + 1

n = sum of ((n-1)!/(n--1-i)!/i!) as i goes from 0 to n-1
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markr
 
  1  
Reply Wed 2 Jul, 2014 08:18 pm
@muna27,
Apparently, order is important.
engineer's example can be simplified to 2^(n-1).
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Stellatyler7
 
  1  
Reply Fri 4 Jul, 2014 05:30 am
@muna27,
Yes, It's Binomial expression.
0 Replies
 
 

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