Chapters

Permutations and Combinations

NCERT Class 11 Mathematics — Chapter 6

The only way to learn mathematics is to do mathematics. — Paul Halmos

6.1 Fundamental Principle of Counting

If one event can occur in mm ways and a second, independently, in nn ways, then both can occur in mnmn ways. This extends to several successive choices.

6.2 Permutations

An arrangement of objects in a definite order is a permutation.

Number of permutations of nn distinct objects taken rr at a time:

nPr=n!(n−r)!,0≤r≤n^nP_r=\frac{n!}{(n-r)!},\qquad 0\le r\le n

n!=n(n−1)⋯1n!=n(n-1)\cdots 1 and 0!=10!=1.

nPn=n!^nP_n=n!.

If pp objects are alike of one kind, qq of another, and so on, the number of distinct permutations of nn objects is

n!p! q! ⋯\frac{n!}{p!\,q!\,\cdots}

6.3 Combinations

A combination is a selection with no regard to order.

nCr=n!r! (n−r)!=nPrr!^nC_r=\frac{n!}{r!\,(n-r)!}=\frac{^nP_r}{r!}

nCr=nCn−r,nCr+nCr−1=n+1Cr^nC_r=^nC_{n-r},\qquad ^nC_r+^nC_{r-1}=^{n+1}C_r

(Pascal’s identity).

Chapter Summary

  • Counting: multiply independent choices
  • nPr=n!/(n−r)!^nP_r=n!/(n-r)! (arrangements)
  • nCr=n!/(r!(n−r)!)^nC_r=n!/(r!(n-r)!) (selections)
  • alike objects: divide by factorials of repeats

Exercises (NCERT)

  • Exercise 6.1 — fundamental principle
  • Exercise 6.2 — permutations
  • Exercise 6.3 — combinations
  • Miscellaneous Exercise