# probability specific people appear in a sample from a group

The Justice League wants to randomly select a group of 7 from among the 40 currently available superheroes/superheroines to investigate a glowing meteorite. What is the probability that Bat Man, Wonder Woman, and Spider Man are all chosen?

Initially, I tried to simplify this problem as follows: Bob, Susan, and Mary are standing near the meteorite. What's the probability that both Bob and Susan are randomly zapped with a mutagenic beam?

I drew out the probability space for this sub-problem and get $P(s,b\ zapped)= \frac{2}{9}$ However, this is with replacement (e.g., Bob can get zapped twice).

   b  s  m
b  .  z  .
s  z  .  .
s  .  .  .


When it is without replacement, I am not sure. Is it $P(s,b\ zapped)=\frac {1}{3}\frac{1}{2}$ or do I need to take into account the different ways people can get zapped, $P(s,b\ zapped)={3\choose 2}\frac {1}{3}\frac{1}{2}$ ?

For the original problem, my initial guess was to try to account for the different ways all three are chosen from seven: $P(B,S,W chosen)= {7 \choose 3}{1 \over 40} {1 \over 39} {1 \over 38} {37 \over 37}{36 \over 36}{ 35 \over 35}{ 34 \over 34}$

However if all three are chosen last then it would seem to be: $P(B,S,W chosen\ last)= {7 \choose 3}{40 \over 40} {39 \over 39} {38 \over 38} {37 \over 37}{1 \over 36}{ 1 \over 35}{ 1 \over 34}$

I also tried to think of treating Bat Man, Wonder Woman, and Spider man as a generic group of 3, but I still seem to be missing the possibility of having those three selected in different orderings, like second, fourth, and last.

$P(B,S,W chosen)= {3 \over 40} {2 \over 39} {1 \over 38} {37 \over 37}{36 \over 36}{ 35 \over 35}{ 34 \over 34}$

-