Math Problem Statement

express the following expressions in the simplest form

Express the following expressions in the simplest form: \(\frac{C^3_2}{C^2_2}\).

Solution

The image shows an expression that appears to involve fractions or binomial coefficients. I'll work on interpreting and simplifying it. The expression seems to be C23C22\frac{C^3_2}{C^2_2}.

Let’s break it down:

  1. C23C^3_2 represents a combination formula, (32)=3!2!(32)!=3221=3\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \cdot 2}{2 \cdot 1} = 3.

  2. C22C^2_2 represents another combination formula, (22)=2!2!(22)!=21211=1\binom{2}{2} = \frac{2!}{2!(2-2)!} = \frac{2 \cdot 1}{2 \cdot 1 \cdot 1} = 1.

Now, the expression becomes: C23C22=31=3.\frac{C^3_2}{C^2_2} = \frac{3}{1} = 3.

Thus, the simplified form of the given expression is 3.

Would you like me to verify the steps or clarify any part? Here are some related questions to expand your understanding:

  1. What is the general formula for combinations?
  2. How is the factorial function used in combination calculations?
  3. How do you compute other forms of binomial coefficients?
  4. Can combinations be applied in real-world problems like probability?
  5. What is the difference between combinations and permutations?

Tip: Always

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Factorial
Simplification of Expressions

Formulas

Combination formula: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Theorems

Basic properties of combinations

Suitable Grade Level

Grades 9-12