# Thread: Sum of distinct product

1. ## Sum of distinct product

Hi,

How do I express (or approximate for larg $\displaystyle m$) the following sum in term of $\displaystyle k,m$ ?

$\displaystyle \sum_{1\leq i_1<\cdots< i_{k} \leq m} i_1i_2\cdots i_k$

Thanks

2. Originally Posted by tah
Hi,

How do I express (or approximate for larg $\displaystyle m$) the following sum in term of $\displaystyle k,m$ ?

$\displaystyle \sum_{1\leq i_1<\cdots< i_{k} \leq m} i_1i_2\cdots i_k$

Thanks
Do you mean you want it written like as another summation? i.e. $\displaystyle \sum_{ i_{k} = 1} ^m i_k$

Or do you want the sequence?

3. Originally Posted by tsal15
Do you mean you want it written like as another summation? i.e. $\displaystyle \sum_{ i_{k} = 1} ^m i_k$

Or do you want the sequence?
I want to eliminate totally the sum symbol by giving an explicit expression in term of k and m or an approximation for m large enough.

4. Originally Posted by tah
I want to eliminate totally the sum symbol by giving an explicit expression in term of k and m or an approximation for m large enough.
Ok. So, for $\displaystyle \sum_{k =1} ^m i_k$,

the sum expression is: $\displaystyle i_1+i_2+...+ i_m$

does this help?

5. And since that simple case does not have a "closed expression", the more general certainly does not.

6. I mean by

$\displaystyle i_1i_2\cdots i_k$

a product, so the expression represent a sum of all possible products of k distinct number between 1 and m. For instance k = 2 and m = 3

$\displaystyle S=1\times 2 + 1\times3 + 2\times 3$

so can't we express it more simply without the sum ?

Thanks for the helps.