mathmath333
  • mathmath333
counting question
Mathematics
katieb
  • katieb
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{How many different sums can be formed by the following }\ ?\hspace{.33em}\\~\\ & \normalsize 5\ \text{dollar},\ 1\ \text{dollar},\ 50\ \text{cents},\ 25\ \text{cents},\ 10\ \text{cents},\ 3\ \text{cents},\ 2\ \text{cents},\ 1\ \text{cent.} \hspace{.33em}\\~\\ \end{align}}\)
welshfella
  • welshfella
I guess if you take 2 at a time its 8C2 right?
welshfella
  • welshfella
the 3 at a time it will be 8C3

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mathmath333
  • mathmath333
yes u can also take 3 ,4, 5 upto 8 at a time
welshfella
  • welshfella
yes - but could there be any duplicates in all these?
mathmath333
  • mathmath333
just need to deduct the double count like example 2 cent and 1 cent taken both are equal to 3 cent taken once
welshfella
  • welshfella
yes exactly
mathmath333
  • mathmath333
^duplication
welshfella
  • welshfella
and of course there is 8C1 singles
mathmath333
  • mathmath333
yes
mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \dbinom{8}{1}+\dbinom{8}{2}+\cdots+\dbinom{8}{8}=2^{8}-1 \hspace{.33em}\\~\\ \end{align}}\)
welshfella
  • welshfella
i think the only duplicates are the ones you mentioned
mathmath333
  • mathmath333
yes but consider 5 dollar +1cent +2 cent =5 dollar+3cent
mathmath333
  • mathmath333
this also duplication
welshfella
  • welshfella
right
welshfella
  • welshfella
and 1 dollar + 3 , 1 dollar + 1 + 2
mathmath333
  • mathmath333
yes all are considered
welshfella
  • welshfella
- also same for 50 , 25, 10 cents
mathmath333
  • mathmath333
yep
welshfella
  • welshfella
only combinations we are concerned with here right? not permutations
welshfella
  • welshfella
- yes - bacuase we are dealing with sums only
mathmath333
  • mathmath333
yes only addition (result) is counted
welshfella
  • welshfella
that identity, = 2^8 - 1 I haven't seen that before.
mathmath333
  • mathmath333
google sum of combinations u will get it
mathmath333
  • mathmath333
https://www.physicsforums.com/threads/sum-of-combinations.83309/
welshfella
  • welshfella
right so how do we go about counting the number of duplicates?
mathmath333
  • mathmath333
yea i m confused about that the main thing
triciaal
  • triciaal

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