Permutations and Combinations

Advanced Applications and Distribution

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Advanced Applications and Distribution

Advanced Applications and Distribution

This lesson covers the following key concepts:

  • Distribution of distinct objects into distinct groups
  • Distribution of identical objects into distinct groups
  • Number of non-negative integer solutions
  • Number of positive integer solutions
  • Geometric applications (lines, triangles, diagonals)
  • Word formation problems with constraints
  • Mixed problems combining P&C concepts

Important Formulas

  • Distributing n distinct into r distinct: rn\text{Distributing } n \text{ distinct into } r \text{ distinct: } r^n
  • Non-negative solutions to x1+x2+...+xr=n:n+r−1Cr−1\text{Non-negative solutions to } x_1 + x_2 + ... + x_r = n: ^{n+r-1}C_{r-1}
  • Positive solutions to x1+x2+...+xr=n:n−1Cr−1\text{Positive solutions to } x_1 + x_2 + ... + x_r = n: ^{n-1}C_{r-1}
  • Lines from n points (no 3 collinear): nC2\text{Lines from } n \text{ points (no 3 collinear): } ^nC_2
  • Triangles from n points (no 3 collinear): nC3\text{Triangles from } n \text{ points (no 3 collinear): } ^nC_3
  • Diagonals of n-gon: nC2−n=n(n−3)2\text{Diagonals of n-gon: } ^nC_2 - n = \frac{n(n-3)}{2}