Partitioning a number!!

Probability Level 4

In how many ways can one write a sum of (at least two) positive integers that add up to $11$ ?

1 solution

Harsh Shrivastava
Oct 19, 2014

Let $n$ be a positive integer.Total number of ways in which we can write a sum of (at least two) positive integers that add up to n is

$2^{n-1}$ $-1$ .

Putting $n = 11$ , we get total number of ways $= \boxed{1023}$

Harsh Can you please provide a proof of formula .

Thanks

A Former Brilliant Member - 6 years, 3 months ago

Consider a number $N$ .

We can write $N$ as $1 + 1 +1 +. . . +1$ (N times).

Now we can place $(,)$ or $+$ between $N$ one's in $2^{N-1}$ ways. But we have to subtract the case when there is no $+$ .

Hence the final formula : $2^{N-1} - 1$ .

Harsh Shrivastava - 6 years, 3 months ago

Thanks Harsh

A Former Brilliant Member - 6 years, 3 months ago

According to this formula Is it that 1+3 and 3+1 are two different sums ???

A Former Brilliant Member - 6 years, 3 months ago

Harsh Shrivastava - 6 years, 3 months ago