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How do you solve combination equations?

Published in Combinatorics 3 mins read

Solving combination equations primarily involves using the combination formula and algebraic manipulation to find unknown values. The combination formula calculates the number of ways to choose r items from a set of n items without regard to order.

Understanding the Combination Formula

The core of solving combination equations is understanding and applying the formula:

*nCr = n! / (r! (n - r)!)**

Where:

  • n is the total number of items in the set.
  • r is the number of items being chosen.
  • ! denotes the factorial (e.g., 5! = 5 4 3 2 1).
  • nCr (also written as nCr or (nr)) represents the number of combinations.

Steps to Solve Combination Equations

  1. Identify the Unknown: Determine what variable you are trying to solve for. It could be n, r, or even the combination value (nCr) itself.

  2. Substitute Known Values: Plug the known values into the combination formula.

  3. Simplify the Factorials: Expand the factorials to identify terms that can be cancelled out, simplifying the equation. For instance, if you have 5!/3!, you can write it as (5 4 3!)/3!, and then cancel out the 3!.

  4. Solve for the Unknown: Use algebraic techniques to isolate and solve for the unknown variable. This might involve cross-multiplication, simplification, or solving a polynomial equation.

Examples

Here are a few examples to illustrate the process:

Example 1: Finding n

Solve for n in the equation: nC2 = 15

  • Apply the Formula: n! / (2! * (n-2)!) = 15
  • Expand and Simplify: n (n-1) (n-2)! / (2 1 (n-2)!) = 15 => n * (n-1) / 2 = 15
  • Solve the Quadratic: n * (n - 1) = 30 => n2 - n - 30 = 0 => (n - 6)(n + 5) = 0
  • Find the Solution: n = 6 or n = -5. Since n must be a positive integer, n = 6.

Example 2: Finding r

Solve for r in the equation: 7Cr = 21

  • Apply the Formula: 7! / (r! * (7-r)!) = 21
  • Recognize Symmetry: Combinations exhibit symmetry. That is, nCr = nC(n-r). In this case, 7C2 = 7C5 = 21.
  • Find the Solution: By calculating, we find that r=2 and r=5 are solutions.

Considerations and Tips

  • Symmetry Property: Remember that nCr = nC(n-r). This can simplify equations. If nCr = nCk, then either r=k or r+k=n.
  • Factorial Properties: Understand how factorials work and simplify.
  • Calculator Usage: Use a calculator with a combination function (nCr) to check your work, especially with larger numbers.
  • Positive Integer Solutions: Remember that n and r must be non-negative integers, and n must be greater than or equal to r. Reject any solutions that don't meet these criteria.

By understanding the combination formula and practicing with different types of equations, you can master the art of solving combination problems.

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