<s>[INST] In how many ways can $5$ students be chosen such that at least one student is chosen from each state? There are $3$ states and $3$ students representing each state. In how many ways can $5$ students be chosen such that at least one student is chosen from each state?
According to me, the answer should be $3C1 \cdot 3C1 \cdot 3C1 \cdot 6C2$, but this answer is wrong.
Can someone please help me?
 [/INST] Your answer over counts.
Suppose the representatives of state $A$ are $a, b, c$, the representatives of state $B$ are $d, e, f$, and the representatives of state $C$ are $g, h, i$.  You count the selection $\{a, b, c, d, g\}$ three times, once for each of the ways we could choose a representative of state $A$. 
$a, d, g, \qquad b, c$
$b, d, g, \qquad a, c$
$c, d, g, \qquad a, b$
and the selection $\{a, b, d, e, g\}$ four times, once for each of the ways we could pick a representative of state $A$ and once for each of the ways we could pick a representative of state $B$.
$a, d, g, \qquad b, e$
$a, e, g, \qquad b, d$
$b, d, g, \qquad a, e$
$b, e, g, \qquad a, d$ 
If we do a direct count, the number of permissible selections is 
$$\binom{3}{3}\binom{3}{1}\binom{3}{1} + \binom{3}{1}\binom{3}{3}\binom{3}{1} + \binom{3}{1}\binom{3}{1}\binom{3}{3} + \binom{3}{2}\binom{3}{2}\binom{3}{1} + \binom{3}{2}\binom{3}{1}\binom{3}{2} + \binom{3}{1}\binom{3}{2}\binom{3}{2} = 3\binom{3}{3}\binom{3}{1}^2 + 3\binom{3}{2}^2\binom{3}{1}$$
Your answer $\binom{3}{1}\binom{3}{1}\binom{3}{1}\binom{6}{2}$ counts each of the first three terms three times and each of the last three terms four times.    Notice that 
$$3 \cdot 3\binom{3}{3}\binom{3}{1}^2 + 4 \cdot 3 \binom{3}{2}^2\binom{3}{1} = \binom{3}{1}^3\binom{6}{2}$$
As RobertZ and Muralidharan indicated, it is easier to subtract the number of cases in which exactly two states are represented from the total number of ways of selecting five representatives from the nine available people.
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