\binom{3}{2} \times (2^5 - 1) = 3 \times 31 = 93

["# Solving \binom{3}{2} × (2⁵ − 1) = 93: A Step-by-Step Breakdown Using Binomial Coefficients and Exponents", "Mathematics often presents elegant expressions that, at first glance, may seem complex—but unlocking their meaning reveals beautiful simplicity. One such expression is:", "[\n\binom{3}{2} \ imes (2^5 - 1) = 93\n]", "This equation combines binomial coefficients and exponential expressions, making it a great example of how foundational math concepts collaborate in concise forms. In this SEO-optimized article, we’ll explore how to evaluate this expression step by step, understand each component, and uncover why this simplified result (93) has practical significance.", "---", "## Breakdown of the Expression", "Let’s interpret the equation clearly:", "- (\binom{3}{2}): This is a binomial coefficient (also called a "combination"), representing the number of ways to choose 2 items from 3.\n- ((2^5 - 1)): This calculates (2) raised to (5), then subtracts (1).\n- The entire expression evaluates to (93), a modest but meaningful number.", "Our goal is to explain how this result arises step by step.", "---", "## Step 1: Evaluate the Binomial Coefficient (\binom{3}{2})", "The combination formula is:", "[\n\binom{n}{k} = \frac{n!}{k!(n-k)!}\n]", "Plugging in (n = 3), (k = 2):", "[\n\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \ imes 2 \ imes 1}{(2 \ imes 1)(1)} = \frac{6}{2 \ imes 1} = 3\n]", "✅ Result: (\binom{3}{2} = 3)", "This step identifies how many ways there are to choose 2 items from 3—there are exactly 3 combinations: if your items are A, B, C, the valid pairs are AB, AC, and BC.", "---", "## Step 2: Calculate (2^5 - 1)", "Following order of operations (PEMDAS/BODMAS), exponentiation comes before subtraction:", "[\n2^5 = 2 \ imes 2 \ imes 2 \ imes 2 \ imes 2 = 32\n]", "Then subtract 1:", "[\n32 - 1 = 31\n]", "✅ Result: (2^5 - 1 = 31)", "This expression, (2^5 - 1), is also significant in computer science—it represents the maximum value of a 5-bit unsigned binary number (from (0) to (31)). It appears in binary arithmetic, masking techniques, and pseudo-random number generation.", "---", "## Step 3: Multiply the Results", "Now multiply the two results:", "[\n\binom{3}{2} \ imes (2^5 - 1) = 3 \ imes 31 = 93\n]", "✅ Final Result: (93)", "This final multiplication demonstrates how combinatorics and exponentiation can combine to produce a clean whole, useful in counting problems or calculating states in algorithmic logic.", "---", "## Why This Equation Matters: Real-World Context", "The expression involves two key mathematical ideas:", "- Combinatorics: (\binom{3}{2}) counts subsets, crucial in probability, statistics, and algorithm analysis.\n- Binary Logic: (2^5 - 1) link to binary systems — efficient for coding, hashing, and networking.", "Together, they explain how discrete structures (sets, bits) interact multiplicatively, forming the backbone of computational methods and data organization.", "---", "## Mathematical Context: General Patterns", "This problem reflects a broader mathematical principle:", "[\n\binom{n}{k} \ imes (a^k - 1)\n]", "when (k = 2) and (a = 2), simplifies neatly when (a) is small, producing values tied to powers of 2 and subset counts.", "Such forms appear in:", "- Hash function design (avoiding collisions over small keys),\n- Error-detection algorithms,\n- Recursive subset generation in programming.", "---", "## SEO Keywords Summary", "To optimize this content naturally, include these high-impact keywords:", "- (\binom{3}{2} \ imes (2^5 - 1))\n- Mathematics problem solution\n- Binomial coefficient explained\n- Power of 2 math\n- Combinatorics in binary systems\n- Subset counting and algebraic evaluation\n- Simplified equation 3 × 31 = 93", "---", "## Conclusion", "The expression (\binom{3}{2} \ imes (2^5 - 1)) elegantly evaluates to (93), showcasing how fundamental concepts—combinations and exponents—merge efficiently. Whether used in teaching combinatorics, explaining binary logic, or streamlining algorithmic logic, this calculation reinforces the power of precise mathematical expression.", "By breaking it down, learners not only compute numbers but understand the structural beauty behind mathematical reasoning—key to mastering advanced topics.", "---", "### Ready to explore more mathematical elegance? Check out related articles:\n- How binomial coefficients shape modern cryptography\n- Exponential growth and binary subtraction in computer science\n- Practical applications of combinatorial counting in everyday tech", "---", "Meta Description for SEO:\nLearn how (\binom{3}{2} \ imes (2^5 - 1)) simplifies to 93 by breaking down binomial coefficients and exponential math. Understand the combinatorial and binary logic behind this elegant equation—perfect for math enthusiasts and computer science learners.", "---", "If you found this breakdown helpful, subscribe to our math education channel and explore more step-by-step solutions to popular expressions!"]