Solution: 2017 Fall Final - 5
Author: Michiel SmidQuestion
Let $n \geq 3$ and $m \geq 3$ be integers. What does
$$
{n \choose 3} + {m \choose 3} + n \cdot {m \choose 2} + m \cdot {n \choose 2}
$$
count?
(a)
The number of subsets having size 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
(b)
The number of subsets having size 2 or 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
(c)
The number of subsets having size at most 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
(d)
The number of subsets having size 2 and 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
Solution
Let’s explain why some are wrong and one is right
- The number of subsets having size 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
This one is right because it lets us pick 3 positive numbers OR 3 negative numbers OR 1 positive number and 2 negative numbers OR 1 negative number and 2 positive numbers - The number of subsets having size at most 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
this one is incorrect because the equation given misses sets of size 1 and 2
For instance, should've said $ \binom{n}{1} + ... $ but they didn't - The number of subsets having size 2 or 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
This one is incorrect because the equation given misses sets of size 1
For instance, should've said $ \binom{n}{1} + ... $ but they didn't - The number of subsets having size 2 and 3 of a set consisting of $n$ positive numbers and $m$ negative numbers.
Uh, a set can't be of both size 2 and 3 at the same time. If I have 2 balls, I don't have 3 balls. If I have 3 balls, I don't have 2 balls