AI Tutor
Imagine your school raffle draws 3 winning tickets out of 30 entries. If all 3 winners receive an identical $25 gift card, does the order your name is called matter?
Not at allβ€”a win is a win. But if the first ticket drawn wins a laptop, the second gets a tablet, and the third gets a sticker, the order of selection changes everything.

The Order Question

A permutation is an arrangement of items where the specific order or sequence matters. A combination is a selection of items where the order does not matter.
To arrange rr items chosen from nn distinct options, there are nn choices for the first slot, (nβˆ’1)(n-1) for the second, down to (nβˆ’r+1)(n-r+1), giving the permutation formula P(n,r)=n!(nβˆ’r)!P(n, r) = \frac{n!}{(n-r)!}.
Because every group of rr items can be rearranged in r!r! identical ways, we divide the permutations by r!r! to eliminate duplicate orderings, giving the combination formula (nr)=n!r!(nβˆ’r)!\binom{n}{r} = \frac{n!}{r!(n-r)!}.
In 1654, French mathematicians Blaise Pascal and Pierre de Fermat formalized these counting rules to solve the 'problem of points,' calculating fair payouts in interrupted games of chance.
How do we turn these counting formulas into exact real-world probabilities?

Probability with Unordered Subsets

For any event with equally likely outcomes, the theoretical probability is P(E)=n(E)n(S)P(E) = \frac{n(E)}{n(S)}, where n(E)n(E) is the number of favorable outcomes and n(S)n(S) is the total size of the sample space.
Suppose a hospital committee randomly selects 4 doctors from a department of 8 surgeons and 6 pediatricians. What is the probability that the committee contains exactly 3 surgeons and 1 pediatrician?