Binomial Theorem, Factorial, Permutations, Combination and Probability Tricks and Formulas

Binomial Theorem

If n is a natural number that is greater than or equal to 2, then according to the binomial theorem:

(x+a)n=ncoxna0+nc1xn1a1+nc2xn2a2+nc3xn3a3++ncnx0an

Here, ncr=n!(nr)!r!

Factorial

n!=1×2×3×...×(n1)×n

n!=n×(n1)!

Permutations

ncrn!(nr)!r!

Combinations

ncr=(nr)!r!n!

Important Properties

ncr=ncnr

nc0+nc1+nc2+nc3++ncn=2n

Partition Rule

Number of ways of distributing n identical things among r persons when each person may get any number of things =n+r1cr1

Probability

  • Probabilityofanevent=NumberoffavourableoutcomesNumberoftotaloutcomes

  • oddsinfavour=NumberoffavourableoutcomesNumberofunfavourableoutcomes

  • oddsagainst=NumberofunfavourableoutcomesNumberoffavourableoutcomes