2019 AIME I Problems/Problem 4

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Problem

A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game (including the possibility of making no substitutions). Find the remainder when $n$ is divided by $1000$.

Solution 1 (Recursion)

There are $0-3$ substitutions. The number of ways to sub any number of times must be multiplied by the previous number. This is defined recursively. The case for $0$ subs is $1$, and the ways to reorganize after $n$ subs is the product of the number of new subs ($12-n$) and the players that can be ejected ($11$). The formula for $n$ subs is then $a_n=11(12-n)a_{n-1}$ with $a_0=1$.

Summing from $0$ to $3$ gives $1+11^2+11^{3}\cdot 10+11^{4}\cdot 10\cdot 9$. Notice that $10+9\cdot11\cdot10=10+990=1000$. Then, rearrange it into $1+11^2+11^3\cdot (10+11\cdot10\cdot9)= 1+11^2+11^3\cdot (1000)$. When taking modulo $1000$, the last term goes away. What is left is $1+11^2=\boxed{122}$.

~BJHHar

Solution 2 (Casework)

We can perform casework. Call the substitution area the "bench".

$\textbf{Case 1}$: No substitutions. There is $1$ way of doing this: leaving everybody on the field.

$\textbf{Case 2}$: One substitution. Choose one player on the field to sub out, and one player on the bench. This corresponds to $11\cdot 11 = 121$.

$\textbf{Case 3}$: Two substitutions. Choose one player on the field to sub out, and one player on the bench. Once again, this is $11\cdot 11$. Now choose one more player on the field to sub out and one player on the bench that was not the original player subbed out. This gives us a total of $11\cdot 11\cdot 11\cdot 10 = 13310\equiv 310 \bmod{1000}$.

$\textbf{Case 4}$: Three substitutions. Using similar logic as $\textbf{Case 3}$, we get $(11\cdot 11)\cdot (11\cdot 10)\cdot (11\cdot 9)$. The resulting number ends in $690$.

Therefore, the answer is $1 + 121 + 310 + 690 \equiv \boxed{122} \pmod{1000}$.

~minor mistake fixed by Prism Melody

Solution 3 (Official MAA)

There is $1$ way of making no substitutions to the starting lineup. If the coach makes exactly one substitution, this can be done in $11^2$ ways. Two substitutions can happen in $11^2\cdot11\cdot 10$ ways. Similarly, three substitutions can happen $11^2\cdot11\cdot10\cdot11\cdot9$ ways. The total number of possibilities is $1+11^2+11^2\cdot11\cdot10+11^2\cdot11\cdot10\cdot11\cdot9=122+11^3(10+990)\equiv 122\pmod{1000}.$

~Shreyas S