The Starting Five

For a roster of 8 players, find the number of possible squads of every size from 0 to 8, and write them in a row. What do you notice? What is the total?

Task 2b: Whatever pattern you noticed, prove it for any roster of \(n\) players in two different ways: once in words, with a coach’s story and no algebra, and once using your factorial formula.

Without a calculator, find \(\binom{50}{48}\) and \(\binom{100}{98}\). Then find \(n\) if \({}^{n}C_{3} = {}^{n}C_{12}\).

Task 2d: The row \(1, 8, 28, 56, 70, 56, 28, 8, 1\) rises and then falls. Find a simplified expression for \({}^{n}C_{r+1} \div {}^{n}C_{r}\). Use it to find where the row peaks for any roster size \(n\), and prove that once the row starts to fall, it never rises again.

Could \({}^{n}C_{a} = {}^{n}C_{b}\) happen with \(a \neq b\) for any reason other than your rule in 2b? Use 2d to settle it, and say what that means for the value of \(n\) you found in 2c.

A roster has 10 players, one of whom is the star, S. (i) How many squads must include S? (ii) How many squads must exclude S? (iii) Compare your two answers with each other, and with the total number of squads of 5 from 10. Explain what you see.

Another roster has 7 forwards and 5 guards. The league requires every squad to include at least 2 guards. How many valid squads are there? Find it, then confirm it with a completely different method.

A club has 11 players. The coach picks a squad of 5, then names one of the five as captain and a different one as vice-captain. How many different outcomes are there? Count them two ways: (i) squad first, then the roles; (ii) roles first, then the rest of the squad. Why must the two answers agree?

A teammate says: “To pick 2 captains and then 3 other players from 10 people, the answer is \(\binom{10}{2} \times \binom{10}{3} = 5400\).” Explain exactly what is wrong, and find the correct number.

Task 3e: Choose \(a\) captains and then \(b\) other players from \(n\) people, with nobody in both groups. Write the number of ways as a product of two choose-numbers, then rewrite it as a single fraction of factorials. Explain why your final expression is unchanged if \(a\) and \(b\) are swapped.

Task 4a: A team of \(k\) players is chosen from \(n\), and one of the \(k\) is named captain. Count the number of possible (team, captain) outcomes in two different ways, and conclude an equation linking \(k\binom{n}{k}\) to a choose-number involving \(n-1\) and \(k-1\). Check it on \(n = 8,\ k = 4\).

Task 4b: A roster of \(n\) players may choose a squad of any size, including no players at all and the whole roster. Count the total number of possible squads in two different ways: (i) by squad size; (ii) by going through the players one at a time. What does this tell you about the sum of a whole row? Compare with your total in 2a.