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## Probability using combinatorics

Current time:0:00Total duration:2:23

# Example: Different ways to pick officers

## Video transcript

A club of nine people wants
to choose a board of three officers, President, Vice
President, and Secretary. Assuming the officers are chosen
at random, what is the probability that the officers
are Marcia for president, Sabita for Vice President,
and Robert for Secretary? So to think about the
probability of Marcia-- so let me write this-- President is
equal to Marcia, or Vice President is equal
to Sabita, and Secretary is equal to Robert. This, right here, is one
possible outcome, one specific outcome. So it's one outcome out of the
total number of outcomes, over the total number of
possibilities. Now what is the total number
of possibilities? Well to think about that, let's
just think about the three positions. You have President, you
have Vice President, and you have Secretary. Now let's just assume that we're
going to fill the slot of President first. We don't
have to do President first, but we're just going
to pick here. So if we're just picking
President first, we haven't assigned anyone to any officers
just yet, so we have nine people to choose from. So there are nine possibilities
here. Now, when we go to selecting our
Vice President, we would have already assigned one
person to the President. So we only have eight
people to pick from. And when we assign our
Secretary, we would've already assigned our President and Vice
President, so we're only going to have seven people
to pick from. So the total permutations here
or the total number of possibilities, or the total
number of ways, to pick President, Vice President, and
Secretary from nine people, is going to be 9 times 8 times 7. Which is, let's see,
9 times 8 is 72. 72 times 7, 2 times 7 is 14, 7
times 7 is 49 plus 1 is 50. So there's 504 possibilities. So to answer the question, the
probability of Marcia being President, Sabita being Vice
President, and Robert being Secretary is 1 over the total
number of possibilities, which is 1 over 504. That's the probability.