Permutations and Combinations Problem

Homework Statement

In how many ways is it possible to select one or more letters from those in INSIPIDITY?

The Attempt at a Solution

My initial impression was that this was a combinations problem and I did not have to take into account the repetitions. This led me to over-count because I had counted combinations like SIPI multiple times as SI2PI3 and SI1PI4 are indistinguishable. The only method I could think of was for each number of selected letters to add up the mutually exclusive scenarios that constituted it, as follows

1 letter selected: no I involved, 1 I involved
2 letters selected: no I's involved, 1 I involved, 2 I's involved
3 letters selected: no I's involved, 1 I involved, 2 I's involved, 3 I's involved etc.

Apparently ((4+1)(1+1)^6) -1 is a much quicker solution, however I am unable to interpret the logic of this solution. Could someone please explain it to me? Also if anyone could offer some tips on how to tackle these problems I would appreciate it, I have found this to be the most difficult subject I have encountered so far, mainly because there doesn't seem to be a procedure to follow like there was for calculus.

The Attempt at a Solution

Related Precalculus Mathematics Homework Help News on Phys.org
mfb
Mentor
You can take (0, 1, 2, 3 or 4) letters "I" -> 4+1 options
You can take (0 or 1) letters "N" -> ...
...
All those choices are independent of each other.

mainly because there doesn't seem to be a procedure to follow like there was for calculus.
There are always many ways to find a solution, the more ways you know and recognize (needs practice!) the easier it is to find a good one.

1 person
Thanks for the help.