Let $A= \{1,2,3,4,5,6,7,8,9,0,20,30,40,50\}$.
1. How many subsets of size $2$ are there?
2.How many subsets are there altogether?
Answer:
1) I think there are $7$ subsets of size two are there, since $14$ elements$/2=7$ and also by grouping them into pairs it shows there are $7$ groups of pairs as follows; $\{\{1,2\}, \{3,4\}, \{5,6\}, \{7,8\}, \{9,0\}, \{20,30\}, \{40,50\}\}$
2) I choose $15$, but I'm not sure.
However, are my answers valid? Hints are much appreciated.
Best Answer
The comments have already very quickly pointed this out, but perhaps I can give an explanation on why they work.
$1$) André Nicolas' comment has already answered that there are $\binom{14}{2}$ subsets of size $2$. This notation is the binomial coefficient. That is, \begin{align}\binom{14}{2} &= \frac{14!}{(14-2)!2!}\\ &= \frac{14 \times 13 \times \color{red}{12} \times \color{red}{11} \times \color{red}{10} \times \color{red}9 \times \color{red}8 \times \color{red}7 \times \color{red}6 \times \color{red}5 \times \color{red}4 \times \color{red}3 \times \color{red}2 \times \color{red}1}{(\color{red}{12} \times \color{red}{11} \times \color{red}{10} \times \color{red}9 \times \color{red}8 \times \color{red}7 \times \color{red}6 \times \color{red}5 \times \color{red}4 \times \color{red}3 \times \color{red}2 \times \color{red}1)(2 \times 1)}\\ &=\frac{14 \times 13}{2}\\ &= 91. \end{align} This is correct because there are $14$ elements in your set, and we wish to choose $2$ of them each time.
$2$) Ethan Hunt's comment has already answered that there are $2^n$ subsets, including the empty set and the entire set itself. To find all subsets of a set, we take the power set of the set. It's easy to prove that the cardinality of the power set is $2^n$, where $n$ is the number of elements of the set. So in your case, there are $2^{14} = 16384$ subsets.