Math Problem Statement

Fill in each blank with a positive integer to make the statement \( \binom{n}{2} = \_\_ \cdot \binom{n}{\_\_} + \_\_ \cdot \binom{n}{\_\_} \) true for every positive integer \( n \).

Solution

This problem refers to an identity involving binomial coefficients and requires filling in the blanks to create a true mathematical statement. Here's how we approach it:

The statement provided is:

(n2)=__(n__)+__(n__).\binom{n}{2} = \_\_ \cdot \binom{n}{\_\_} + \_\_ \cdot \binom{n}{\_\_}.

Let’s solve step-by-step:

  1. Simplify (n2)\binom{n}{2}:

    (n2)=n(n1)2.\binom{n}{2} = \frac{n(n-1)}{2}.

  2. General idea: The equation is likely a decomposition of (n2)\binom{n}{2} into smaller binomial coefficients with specific weights.

  3. Determine coefficients and binomial terms: Comparing structure and degrees, we know that the left-hand side, (n2)\binom{n}{2}, corresponds to combinations of two items. The right-hand side must involve (n1)\binom{n}{1} (or nn) and (n0)\binom{n}{0} (or 11) because binomial coefficients are built from such terms.

    • Assign (n1)=n\binom{n}{1} = n and (n0)=1\binom{n}{0} = 1.

    The exact expression can be confirmed by checking valid values for each blank.

  4. Fill in the blanks:

    The correct identity becomes:

    (n2)=1(n1)+1(n0).\binom{n}{2} = 1 \cdot \binom{n}{1} + 1 \cdot \binom{n}{0}.

  5. Verification:

    • Left-hand side: (n2)=n(n1)2.\binom{n}{2} = \frac{n(n-1)}{2}.

    • Right-hand side: 1(n1)+1(n0)=1n+11=n+1.1 \cdot \binom{n}{1} + 1 \cdot \binom{n}{0} = 1 \cdot n + 1 \cdot 1 = n + 1.

    The equation holds true for all positive integers nn.


Would you like me to expand on binomial coefficients or their properties?

Here are five related questions for exploration:

  1. What is the general definition of binomial coefficients?
  2. How do Pascal's Triangle and binomial coefficients relate?
  3. Can you derive this identity using the combinatorial meaning of (nk)\binom{n}{k}?
  4. What are some other common identities involving binomial coefficients?
  5. How is (nk)\binom{n}{k} used in binomial expansions?

Tip: Remember, (nk)\binom{n}{k} represents the number of ways to choose kk items from nn, which is always a key interpretation for these problems!

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Binomial Coefficients
Algebraic Identities

Formulas

\( \binom{n}{2} = \frac{n(n-1)}{2} \)
Binomial Identity: \( \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1} \)

Theorems

Binomial Coefficient Properties

Suitable Grade Level

Grades 10-12