2022 AIME II Problems/Problem 10

Problem

Find the remainder when$$\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}$$is divided by $1000$.

~ pi_is_3.14

Solution 1

To solve this problem, we need to use the following result:

$$\sum_{i=n}^m \binom{i}{k} = \binom{m+1}{k+1} - \binom{n}{k+1} .$$

Now, we use this result to solve this problem.

We have \begin{align*} \sum_{i=3}^{40} \binom{\binom{i}{2}}{2} & = \sum_{i=3}^{40} \binom{\frac{i \left( i - 1 \right)}{2}}{2} \\ & = \sum_{i=3}^{40} \frac{\frac{i \left( i - 1 \right)}{2} \left( \frac{i \left( i - 1 \right)}{2}- 1 \right)}{2} \\ & = \frac{1}{8} \sum_{i=3}^{40} i \left( i - 1 \right) \left( i \left( i - 1 \right) - 2 \right) \\ & = \frac{1}{8} \sum_{i=3}^{40} i \left( i - 1 \right) \left( \left( i - 2 \right) \left( i - 3 \right) + 4 \left( i - 2 \right) \right) \\ & = 3 \left( \sum_{i=3}^{40} \binom{i}{4} + \sum_{i=3}^{40} \binom{i}{3} \right) \\ & = 3 \left( \binom{41}{5} - \binom{3}{5} + \binom{41}{4} - \binom{3}{4} \right) \\ & = 3 \left( \binom{41}{5} + \binom{41}{4} \right) \\ & = 3 \cdot \frac{41 \cdot 40 \cdot 39 \cdot 38}{5!} \left( 37 + 5 \right) \\ & = 3 \cdot 41 \cdot 13 \cdot 38 \cdot 42 \\ & = 38 \cdot 39 \cdot 41 \cdot 42 \\ & = \left( 40 - 2 \right) \left( 40 - 1 \right) \left( 40 + 1 \right) \left( 40 + 2 \right) \\ & = \left( 40^2 - 2^2 \right) \left( 40^2 - 1^2 \right) \\ & = \left( 40^2 - 4 \right) \left( 40^2 - 1 \right) \\ & = 40^4 - 40^2 \cdot 5 + 4 . \end{align*}

Therefore, modulo 1000, $\sum_{i=3}^{40} \binom{\binom{i}{2}}{2} \equiv \boxed{\textbf{(004) }}$.

~Steven Chen (www.professorchenedu.com)

Solution 2 (similar to solution 1)

Doing simple algebra calculation will give the following equation: \begin{align*} \binom{\binom{n}{2}}{2} = \frac{\frac{n(n-1)}{2} \cdot (\frac{n(n-1)}{2}-1)}{2} \\ = \frac{n(n-1)(n^2-n-2)}{8} \\ = \frac{(n+1)n(n-1)(n-2)}{8} \\ = \frac{(n+1)!}{8\cdot (n-3)!} = 3 \cdot \frac{(n+1)!}{4!\cdot (n-3)!} \\ = 3 \binom{n+1}{4} \end{align*}

Next, by using Hockey-Stick Identity, we have: $$3 \cdot \sum_{i=3}^{40} \binom{n+1}{4} = 3 \binom{42}{5} = 42 \cdot 41 \cdot 39 \cdot 38$$ $$=(40^2-2^2)(40^2-1^2) \equiv \boxed{004} (mod 1000)$$

~DSAERF-CALMIT (https://binaryphi.site)

Solution 3

Since $40$ seems like a completely arbitrary number, we can use Engineer's Induction by listing out the first few sums. These are, in the order of how many terms there are starting from $1$ term: $3$, $18$, $63$, $168$, $378$, and $756$. Notice that these are just $3 \cdot \dbinom50$, $3 \cdot \dbinom61$, $3 \cdot \dbinom72$, $3 \cdot \dbinom83$, $3 \cdot \dbinom94$, $3 \cdot \dbinom{10}5$. It's clear that this pattern continues up to $38$ terms, noticing that the "indexing" starts with $\dbinom32$ instead of $\dbinom12$. Thus, the value of the sum is $3 \cdot \dbinom{42}{37}=2552004 \equiv \boxed{\textbf{004}} \pmod{1000}$.

~A1001

Solution 4

As in solution 1, obtain $\sum_{i=3}^{40} \binom{\binom{i}{2}}{2} = \frac{1}{8} \sum_{i=3}^{40} i^4-2i^3-i^2+2i.$ Write this as $$\frac{1}{8}\left(\sum_{i=3}^{40} i^4 - 2\sum_{i=3}^{40}i^3 - \sum_{i=3}^{40}i^2 + 2\sum_{i=3}^{40}i\right).$$

We can safely write this expression as $\frac{1}{8}\left(\sum_{i=1}^{40} i^4 - 2\sum_{i=1}^{40}i^3 - \sum_{i=1}^{40}i^2 + 2\sum_{i=1}^{40}i\right)$, since plugging $i=1$ and $i=2$ into $i^4-2i^3-i^2+2i$ both equal $0,$ meaning they won't contribute to the sum.

Use the sum of powers formulae. We obtain $$\frac{1}{8}\left(\frac{i(i+1)(2i+1)(3i^2+3i-1)}{30} - \frac{i^2(i+1)^2}{2} - \frac{i(i+1)(2i+1)}{6} + i(i+1)\right) \text{ where i = 40.}$$

We can factor the following expression as $\frac{1}{8}\left(\frac{i(i+1)(2i+1)(3i^2+3i-6)}{30} - \frac{i(i+1)}{2} (i(i+1)-2)\right),$ and simplifying, we have $$\sum_{i=3}^{40} \binom{\binom{i}{2}}{2} = \frac{i(i+1)(2i+1)(i^2+i-2)}{80}-\frac{i^2(i+1)^2-2i(i+1)}{16} \text{ where i = 40.}$$

Substituting $i=40$ and simplifying gets $41\cdot 81\cdot 819 - 5\cdot 41\cdot 819,$ so we would like to find $819\cdot 76\cdot 41 \pmod{1000}.$ To do this, get $819\cdot 76\equiv 244 \pmod{1000}.$ Next, $244\cdot 41 \equiv \boxed{004} \pmod{1000}.$

-sirswagger21

Solution 5 (Telescoping)

\begin{align*} \sum_{i=3}^{40} \binom{\binom{i}{2}}{2} & = \frac{1}{8}\sum_{i=3}^{40} (i-2)(i-1)i(i+1) \\ & = \frac{1}{40}\sum_{i=3}^{40}[(i-2)(i-1)i(i+1)(i+2)-(i-3)(i-2)(i-1)i(i+1)] \\ & = \frac{38\cdot39\cdot40\cdot41\cdot42-0}{40}\\ & \equiv \boxed{004}\pmod{1000}\ \end{align*} For the last step, see Solution 1.

~qyang