Combinatoric in Accounting

Prove:
$latex \displaystyle\sum_{n=2}^{n}{_n}C_r = 2^n-1-n$
$latex \displaystyle\sum_{n=2}^{n}{_n}C_r 2^r= 3^n-1-2n $
Note:
$latex {_4}C_2=\frac {4.3}{2!}$
Proof:
1.
$latex \displaystyle\sum_{n=2}^{n}{_n}C_r $ =(1+1)ⁿ -1- $latex \displaystyle {_n}C_1 =2^n-1-n$
2.
$latex \displaystyle\sum_{n=2}^{n}{_n}C_r 2^{r}$=(1+2)ⁿ -1 - $latex \displaystyle {_n}C_1 .2^{1}= 3^n-1-2n$

An accounting transaction = Debit p accounts + Credit q accounts (p, q ≥ 1)

In a company with total n accounts,
Prove: there are $latex 3^n -2^{n+1} + 2$ transactions.

Proof:
Let the Set of all accounts T = {a1, a2, ..., an}
aj = account with value '+' (credit), or '-' (debit), or 0 (nil)
1. Trivial transaction: To= {0, 0,.....0} = 1 way
2. Choose r accounts from n = $latex \displaystyle {_n}C_r $
3. Slot '+' or '-' in these r accounts = $latex 2^r $ways
exclude 2 impossible all '+', '-' transactions = $latex 2^r- 2$ ways
4.
Let T1 = $latex \displaystyle\sum_{n=2}^{n}{_n}C_r .(2^r-2)$
Total transaction = To + T1
= $latex \displaystyle 1+ \sum_{n=2}^{n}{_n}C_r .(2^r-2)$
{Apply previous results}
= $latex \displaystyle 1+ \sum_{n=2}^{n}{_n}C_r .2^r -2.\sum_{n=2}^{n}{_n}C_r$
= $latex 1+3^n-1-2n -2(2^n-1-n) $
= $latex 3^n -2^{n+1} + 2$  [QED]