How many ordered pairs $(A,B)$, where $A ,B$ are subsets of $\{1,2,3,4,5\}$, are there if |$A|+|B|=4$?
How many ordered pairs $(A,B)$, where $A , B$ are subsets of $\{1,2,3,4,5\}$, are there if $|A|+|B|=4$?
Solution 1:
Hint: There are five cases. Count each separately, then sum.
(a) $|A|=0, |B|=4$
(b) $|A|=1, |B|=3$
(c) $|A|=2, |B|=2$
(d) $|A|=3, |B|=1$
(e) $|A|=4, |B|=0$
Solution 2:
It's the same as the number of ordered pairs $(A,B)$ where $A$ is a subset of $\{1,2,3,4,5\}$ and $B$ is a subset of $\{6,7,8,9,10\}$ and $|A|+|B|=4$, which is the same as the number of sets $X$ such that $X$ is a subset of $\{1,2,3,4,5,6,7,8,9,10\}$ and $|X|=4$, which is $\binom{10}4=210$.