no of ways to fill a row (1xN grid) with a set of 1D bars? Given a row of length $N$, and a set of 1D bars having lengths $A[1...M]$, how many ways I can fill the row? 
A is an integer array, 
the bars are having dimensions $\{ 1\times A_1, 1\times A_2, 1\times A_3,..., 1\times A_M \}$ 
The row can also be considered as $1\times N$ grid.
2 Bars with equal length has to be considered as $2$ distinct bar not $1$.
Bars are same from either direction, so just by reversing bars there wont be any new arrangement
I thought solving the problem by converting it into : 
Let, Each space is an element of $1\times 1$, and all spaces are indistinguishable and similar.
No of space $K=N-sum(A_1,A_2,...,A_M)$
Now the answer will be 
No of arrangements of $M$ distinct elements and $K$ similar elements.
But again with no hope as could not find any solution for both the problems.
**Examples : **
$N=3, A=\{1,1\} $ 
$ M=2, K=1$
$Ans = 6 $
$N=3, A=\{1,2\} $ 
$ M=2, K=0$
$Ans = 2 $
$N=4, A=\{1,2\} $ 
$ M=2, K=1$
$Ans = 6 $
$N=3, A=\{2\} $ 
$ M=1, K=1$
$Ans = 2 $
$N=10, A=\{7\} $ 
$ M=1, K=3$
$Ans = 4 $

I found one easy approach : 
Convert the problem further to No of distinct permutation of M distinct letters(the bars) and K letters(the spaces) of same type 
Here Total letters $n=M+K,$ 
Total letters of 1 category  $n_i$ $ = K $ 
Hence total distinct permutation = $\frac{(\sum_i n_i)!}{\prod_i n_i!}$ 
 = $\frac{(M+K)!}{K}$
I found the formula here  .
