You’ve just been hired as head coach of a basketball club. Every week the league office needs to know how many different team sheets you could possibly submit. There are different forms with different rules, and the office has been burned by formulas that only work for small rosters. This time every formula must be justified for any roster size, not just checked on a few.
Roster: A, B, C, D, E. A lineup sheet names a different player for each of 3 positions (point guard, wing, center). How many lineup sheets?
A squad sheet lists only which 3 players make the team. How many squad sheets? Find it by listing.
\({}^{n}P_{r}\) = number of lineups (order matters). \({}^{n}C_{r}\) or \(\binom{n}{r}\) = number of squads (order does not matter). Work them out by hand.
Compare your answers: what is the relationship, and why? Test on 6 players (choose 3, then 4) and write a general formula for \({}^{n}C_{r}\).
Prove your relationship between \({}^{n}P_{r}\) and \({}^{n}C_{r}\) for any \(n\) and \(r\).
No calculator, show the cancellation: \({}^{8}P_{3}\), \(\binom{10}{3}\), \(\binom{12}{4}\).
What should \(\binom{n}{0}\) and \(\binom{n}{n}\) equal? Explain, and check against your formula. What does this say about \(0!\)?
Roster of 8. Find the number of squads of every size from 0 to 8, in a row. What do you notice? What is the total?
Prove the pattern for any \(n\) in two ways: in words (no algebra), and with your factorial formula.
No calculator: \(\binom{50}{48}\) and \(\binom{100}{98}\). Find \(n\) if \({}^{n}C_{3} = {}^{n}C_{12}\).
The row \(1, 8, 28, 56, 70, 56, 28, 8, 1\) rises then falls. Simplify \({}^{n}C_{r+1} \div {}^{n}C_{r}\). Where does the row peak for any \(n\)? Prove it never rises again once it starts to fall.
Can \({}^{n}C_{a} = {}^{n}C_{b}\) with \(a \neq b\) for any reason other than your rule in 2b? Use 2d to settle it.
Unless stated otherwise, every squad has exactly 5 players.
10 players, one of them the star S. (i) How many squads include S? (ii) How many exclude S? (iii) Compare with each other and with all squads of 5 from 10.
7 forwards and 5 guards. Every squad needs at least 2 guards. How many valid squads? Confirm with a completely different method.
11 players. Pick a squad of 5, then a captain and a different vice-captain from it. How many outcomes? Count two ways (squad first; roles first). Why must they agree?
A teammate says: 2 captains, then 3 others from 10, is \(\binom{10}{2}\times\binom{10}{3} = 5400\). What is wrong? Find the correct number.
Choose \(a\) captains then \(b\) others from \(n\) people, no overlap. Write as a product of two choose-numbers, then as one factorial fraction. Why is it unchanged if \(a\) and \(b\) swap?
A team of \(k\) from \(n\), one named captain. Count (team, captain) outcomes two ways and conclude an equation linking \(k\binom{n}{k}\) to a choose-number with \(n-1\) and \(k-1\). Check on \(n=8,\ k=4\).
A roster of \(n\) may pick a squad of any size (none, or everyone). Count all possible squads two ways: by size, and player by player. What does this say about the sum of a row? Compare with 2a.
(i) Every squad of \(r\) either includes the star S or not. Prove \(\binom{n}{r} = \binom{n-1}{r-1} + \binom{n-1}{r}\) and say how it relates to 3a. (ii) Prove that for \(n \ge 1\), exactly half of the \(2^n\) squads have an even number of players.