how many ways can we choose a committee of 6 animals if there must be at least 3 pigs?

Question

Can anyone help me this problem?

In a group of $2$ cats, $3$ dogs, and $10$ pigs, in how many ways can we choose a committee of $6$ animals if there must be at least $3$ pigs? Assume each kind of animal is identical.

Answer 1

This is equivalent to the number of integral solutions to the system

$$\begin{cases} x_1+x_2+x_3=6\\0\leq x_1\leq 2\\ 0\leq x_2\leq 3\\ 3\leq x_3\color{grey}{\leq 10}\end{cases}$$

The upper limit on $x_3$ is irrelevant since it is impossible for us to select more pigs than we have in our group of six.

Let $S$ be the set of solutions where we don't care about any of the upper bounds. Let $A_1$ be the set of solutions where (at the very least) the first upperbound condition is violated (and possibly the second one too). Let $A_2$ be the set of solutions where the second upperbound condition is violated.

We wish to count $|S\setminus (A_1\cup A_2)| = |S|-|A_1|-|A_2|+|A_1\cap A_2|$

What does it mean for an upper bound condition? Well, $A_1$ is when the condition $x_1\leq 2$ is false, i.e. $x_1>2$ i.e. $3\leq x_1$. That is to say $|A_1|$ is the number of solutions to

$$\begin{cases}x_1+x_2+x_3=6\\ 3\leq x_1\\ 0\leq x_2\\ 3\leq x_3\end{cases}$$

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Remember to continue calculations that the number of solutions to the system

$$\begin{cases}x_1+x_2+\dots+x_r=n\\ 0\leq x_i~~\forall i\end{cases}$$

is seen to be $\binom{n+r-1}{r-1}$ via stars and bars.

Answer 2

Three seats on the committee are for pigs only, and the other three are at-large. Let's count the ways to fill the at-large seats, since there's exactly one way to fill the reserved seats. Each way corresponds to a monomial of degree $3$ in this product:

$$ (1 + c + c^2)(1 + d + d^2 + d^3)(1 + p + p^2 + p^3) \enspace, $$

where $c^id^jp^k$ represents a committee with $i$ cats, $j$ dogs, and $k$ pigs. There are nine such terms with non-negative exponents and $i \leq 2$:

$$ p^3, dp^2, d^2p, d^3, cp^2, cdp, cd^2, c^2p, c^2d \enspace. $$

Hence there are nine ways to form the committee.