What is the value of ${ }^n{\mathrm{C}}_0$?

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What is the value of nC₀?

In combinatorics, nC₀ represents the number of ways to choose 0 items from a set of n items, which mathematically is expressed as:

nC₀ = ^n!/_0!(n-0)!

Step-by-Step Explanation

Understanding the Formula:
The formula for combinations is given as:

nC_r = ^n!/_r!(n-r)!

Here:

• n! (n factorial) is the product of all positive integers up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
• r! is the factorial of the number of items selected.
• (n-r)! is the factorial of the difference between the total items (n) and the items selected (r).

Applying to nC₀:
When r = 0, the formula simplifies to:

nC₀ = ^n!/_0!(n-0)!

Special Property of Factorials:
By definition, 0! = 1. This is a key mathematical convention used across various combinatorial calculations.

Final Calculation:
Substituting 0! = 1 into the formula:

nC₀ = ^n!/_{1 × n!} = 1

Conceptual Insight: Why is nC₀ Always 1?

Choosing 0 items from n items means you are selecting nothing. There is exactly one way to do this — choose no items. Therefore, regardless of the value of n, the value of nC₀ is always 1.

Practical Example:

If you have a set of 5 items: {A, B, C, D, E}, the number of ways to choose 0 items is exactly 1 — the empty set {}.