The equation provided is a combinatorial identity involving binomial coefficients. The goal is to fill in the blanks to make the equation true for every positive integer $n$.

The given equation can be written as:

$\binom{n}{2} = \text{(blank)} \cdot \binom{n}{\text{(blank)}} + \text{(blank)} \cdot \binom{n}{\text{(blank)}}.$

We need to identify the appropriate integers for the blanks.

Step-by-step explanation:

1. The left-hand side, $\binom{n}{2}$, represents the number of ways to choose 2 objects from $n$ objects.

2. The right-hand side should decompose $\binom{n}{2}$ into a weighted sum of two binomial coefficients.

3. A standard decomposition for $\binom{n}{2}$ is: $\binom{n}{2} = 1 \cdot \binom{n}{1} + 1 \cdot \binom{n}{0}.$

   Explanation:

   • $\binom{n}{1}$ counts the ways to select 1 object, multiplied by a coefficient of 1.
   • $\binom{n}{0}$ counts the ways to select no objects (always equal to 1), multiplied by a coefficient of 1.

4. Substituting into the blanks, the complete equation becomes: $\binom{n}{2} = 1 \cdot \binom{n}{1} + 1 \cdot \binom{n}{0}.$

Final Answer:

The blanks should be filled as:

• First blank: $1$,
• Second blank: $1$,
• Third blank: $1$,
• Fourth blank: $0$.

Let me know if you need further details or explanations!

Here are 5 related questions to expand your understanding:

1. What is the formula for the binomial coefficient $\binom{n}{k}$, and how is it derived?
2. Can you explain why $\binom{n}{0} = 1$ for any positive integer $n$?
3. How does this identity relate to Pascal’s triangle?
4. Can this identity be extended to cases where we decompose $\binom{n}{3}$?
5. What is the combinatorial interpretation of multiplying a binomial coefficient by a scalar?

Tip: Understanding binomial coefficients often involves recognizing their relationship with combinations and Pascal’s triangle. This can help in visualizing and verifying identities!