## mathmath333 one year ago counting question

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{How many different sums can be formed by the following }\ ?\hspace{.33em}\\~\\ & \normalsize 5\ \text{dollar},\ 1\ \text{dollar},\ 50\ \text{cents},\ 25\ \text{cents},\ 10\ \text{cents},\ 3\ \text{cents},\ 2\ \text{cents},\ 1\ \text{cent.} \hspace{.33em}\\~\\ \end{align}}

2. welshfella

I guess if you take 2 at a time its 8C2 right?

3. welshfella

the 3 at a time it will be 8C3

4. mathmath333

yes u can also take 3 ,4, 5 upto 8 at a time

5. welshfella

yes - but could there be any duplicates in all these?

6. mathmath333

just need to deduct the double count like example 2 cent and 1 cent taken both are equal to 3 cent taken once

7. welshfella

yes exactly

8. mathmath333

^duplication

9. welshfella

and of course there is 8C1 singles

10. mathmath333

yes

11. mathmath333

\large \color{black}{\begin{align} & \dbinom{8}{1}+\dbinom{8}{2}+\cdots+\dbinom{8}{8}=2^{8}-1 \hspace{.33em}\\~\\ \end{align}}

12. welshfella

i think the only duplicates are the ones you mentioned

13. mathmath333

yes but consider 5 dollar +1cent +2 cent =5 dollar+3cent

14. mathmath333

this also duplication

15. welshfella

right

16. welshfella

and 1 dollar + 3 , 1 dollar + 1 + 2

17. mathmath333

yes all are considered

18. welshfella

- also same for 50 , 25, 10 cents

19. mathmath333

yep

20. welshfella

only combinations we are concerned with here right? not permutations

21. welshfella

- yes - bacuase we are dealing with sums only

22. mathmath333

yes only addition (result) is counted

23. welshfella

that identity, = 2^8 - 1 I haven't seen that before.

24. mathmath333

google sum of combinations u will get it

25. mathmath333
26. welshfella

right so how do we go about counting the number of duplicates?

27. mathmath333

yea i m confused about that the main thing

28. triciaal

@phi