We must choose 2 qubits out of 5 to be in state \( |1\rangle \), and the remaining 3 automatically in \( |0\rangle \). Since the qubits are distinguishable, the number of configurations is simply the number of ways to choose 2 positions out of 5:

["Title: How Many Ways Can We Choose 2 Qubits in State ( |1\rangle ) Out of 5? A Simple Combinatorics Explained", "---", "Meta Description: Learn how to calculate the number of ways to select 2 out of 5 distinguishable qubits to be in state ( |1\rangle ), with the rest in ( |0\rangle ), using basic combinatorics. Perfect for quantum computing enthusiasts and students.", "---", "Introduction\nIn quantum computing, qubits represent the fundamental units of information, existing in superpositions of states like ( |0\rangle ) and ( |1\rangle ). However, sometimes we need deterministic control—such as selecting exactly 2 qubits to be in the excited state ( |1\rangle ), while the remaining 3 stay in ( |0\rangle ). When the qubits are distinguishable (i.e., labeled or uniquely identifiable), determining the number of such configurations relies entirely on combinatorics—specifically, the number of ways to choose 2 positions out of 5.", "This simple yet powerful concept lies at the heart of many quantum algorithms and error correction schemes, where precise state preparation is essential.", "---", "### Understanding the Quantum Configuration", "Imagine you have 5 distinct qubits labeled Q₁, Q₂, Q₃, Q₄, and Q₅. To prepare a target quantum state, you want exactly 2 of these qubits to be in ( |1\rangle ), and the other 3 to remain in ( |0\rangle ). Because the qubits are distinguishable, the configuration where qubit 1 and 2 are ( |1\rangle ), and the rest ( |0\rangle ), is different from the configuration where qubits 3 and 5 are the ones set to ( |1\rangle ).", "---", "### The Combinatorics Behind the Counting", "Since the order of selection does not matter in this setup—choosing Q₁ and Q₂ is the same as selecting them in reverse—we use the mathematical tool of combinations.", "The number of ways to choose 2 qubits from 5 is given by the binomial coefficient:", "[\n\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \ imes 4}{2 \ imes 1} = 10\n]", "So, there are exactly 10 distinct configurations where exactly two out of five distinguishable qubits are in state ( |1\rangle ), and the other three remain in ( |0\rangle ).", "---", "### Why Distinguishability Matters", "If the qubits were indistinguishable (e.g., uncounted particles in a bosonic system), the counting method would differ—focusing on multisets or symmetrical states. But in this case, the qubits are uniquely labeled. This distinction increases the configurational space dramatically compared to systems with no labeling.", "Each of these 10 configurations represents a valid basis state in a 5-qubit Hilbert space, useful in quantum information encoding, quantum memory initialization, and fault-tolerant computing.", "---", "### Practical Implications", "Understanding how many such configurations exist is crucial for:\n- Designing quantum communication protocols\n- Optimizing quantum circuits for specific state preparation\n- Analyzing error resilience in quantum memory", "For example, if a protocol requires preparing exactly 2 qubits in ( |1\rangle ), knowing there are 10 possible distinct setups helps estimate probabilities, statistical likelihoods, and resource allocation.", "---", "### Final Note", "Selecting 2 qubits out of 5 to be ( |1\rangle ) is a foundational combinatorial task deeply linked to quantum state engineering. With just 10 unique arrangements among distinguishable qubits, precise control becomes both manageable and powerful—key to advancing quantum technologies.", "---", "Summary\n- You have 5 distinguishable qubits.\n- Choose exactly 2 to be in state ( |1\rangle ), rest in ( |0\rangle ).\n- Number of configurations = number of combinations:\n [\n \binom{5}{2} = 10\n ]\n- This counting principle underpins many quantum computing applications.", "---", "Related Topics:\n- Quantum superposition and basis states\n- Combinatorics in quantum information\n- Distinguishable vs. indistinguishable quantum systems\n- Quantum state preparation and measurement", "---", "Keywords:\nquantum computing, qubit states, ( |0\rangle ), ( |1\rangle ), combinations, binomial coefficient, probabilistic quantum states, quantum information science", "---", "Secure internal link: Learn more about quantum error correction and quantum memory initialization → [Read more on quantum state stability].", "---", "Author Bio:\nUltimate Quantum — your guide to mastering the fundamentals and frontiers of quantum computing, one qubit at a time.", "---", "> Count your choices wisely—because in quantum, every two out of five matter."]