Number of subsets with conditions

Question

How can I calculate number of subsets with conditions of their cardinality? for example:

Let $A=\{1,2,3,4,5,6\}$

How many subsets of $A$ there are with at least $3$ elements?

How many subsets of $A$ there are with at most $3$ elements?

Answer 1

Noting that "at least" means "greater than or equal to" and "at most" means "less than or equal to," we get that: \begin{align*} \text{# of Subsets of size at least $3$} & = \text{# of Subsets of Length $3$} + \text{# of Subsets of Length $4$} + \text{# of Subsets of Length $5$} + \text{# of Subsets of Length $6$} \\ & = {6 \choose 3} + {6 \choose 4} + {6 \choose 5} + {6 \choose 6} \\ & = 20 + 15 + 6 + 1 \\ & = 42 \end{align*} \begin{align*} \text{# of Subsets of size at most $3$} & = \text{# of Subsets of Length $0$} + \text{# of Subsets of Length $1$} + \text{# of Subsets of Length $2$} + \text{# of Subsets of Length $3$} \\ & = {6 \choose 0} + {6 \choose 1} + {6 \choose 2} + {6 \choose 3} \\ & = 1 + 6 + 15 + 20 \\ & = 42 \end{align*}

Answer 2

In the first part, you want to choose subsets that have at least 3 elements in the subset.

You can write it as choosing $i$ elements from the set where $i\ge 3$ $$\sum_{i=3}^6\binom{6}{i}=42$$

similarly for the second part $i\le3$ $$\sum_{i=0}^3\binom{6}{i}=42$$

Answer 3

The number of subsets with $k$ elements from a set with $n$ elements is $n\choose k$. This is because a subset is uniquely identifiable by its elements, So if you have $k$ elements, then you have $n\choose k$ methods to choose those elements, and each one of those methods corresponds to a different subset. Notice that this does not leave out any $k$-element subset.

So for at least three, you have $k$ from $3$ to $6$. This is ${6\choose3} + {6\choose4}+{6\choose5}+{6\choose6}=42$.

For the second it is ${6\choose0} + {6\choose1}+{6\choose2}+{6\choose3}=42$.