What is the total number of combinations of 5 items together when there are no duplicates?
Solution 1:
Let $$nCr=\binom{n}{r}=\frac{n!}{k!(n-k)!}$$ Remember that the $\frac{n!}{(n-k)!}$ gives all the permutations and the $k!$ in the denominator is what disregards duplicates.
Now; you want all the ways you can choose $$(1 \text{ category from } 5) + (2 \text{ category from } 5) + \dots + (5 \text{ category from } 5)$$ i.e. $$\binom{5}{1}+\binom{5}{2}+\binom{5}{3}+\binom{5}{4}+\binom{5}{5}=2^5-1=31$$ Note that this follows from the fact that $$(1+1)^n=\binom{n}{0}+\binom{n}{1}+\cdots+\binom{n}{n-1}+\binom{n}{n}=2^n$$ Subtracting $\binom{n}{0}$ from both sides gives us $$\binom{n}{1}+\cdots+\binom{n}{n-1}+\binom{n}{n}=2^n-\binom{n}{0}$$ But since $\binom{n}{0}=1,\forall n\in\mathbb{N}$ we have that $$\binom{n}{1}+\cdots+\binom{n}{n-1}+\binom{n}{n}=2^n-1$$ When $n=5$ we thus get the above answer.
Addendum: To address your concern that there seems to be more than $31$ combinations, here is a list of all the possibilities: $$\begin{array}{|c|c|c|c|c|c|c|} & 1 \text{ category} & 2 \text{ categories} & 3 \text{ categories} & 4 \text{ categories} & 5 \text{ categories} & \text{Sum}\\ \hline & A & AB & ABC & ABCD & ABCDE\\ \hline & B & AC & ABD & ABCE \\ \hline & C & AD & ABE & ABDE \\ \hline & D & AE & ACD & ACDE \\ \hline & E & BC & ACE & BCDE \\ \hline & & BD & ADE \\ \hline & & BE & BCD \\ \hline & & CD & BCE \\ \hline & & CE & BDE \\ \hline & & DE & CDE \\ \hline \text{Total} & 5 & 10 & 10 & 5 & 1 & 31 \\ \hline \end{array}$$
Solution 2:
There are $\binom{5}{1}$ combinations with 1 item, $\binom{5}{2}$ combinations with $2$ items,...
So, you want : $$\binom{5}{1}+\cdots+\binom{5}{5}=\left(\binom{5}{0}+\cdots+\binom{5}{5}\right)-1=2^5-1=31$$
I used that $$\sum_{k=0}^n\binom{n}{k}=(1+1)^n=2^n$$
Solution 3:
Think about this from another angle. You want some number of these five categories without repetition.
Well each category is either chosen by you or not chosen by you. Each such choice bears no relationship with the other choices.
Thus there are $2^5 = 32$ possibilities. However, you are not counting the choice of none of the five categories, so we subtract $1$ to get $31$ possibilities.