Permutations

Multiple Principle or Fundamental Principle of Counting: If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m x n.Addition principle: If there are two mutually exclusive events, such that they can be occurred independently in m and n ways respectively, then either of the two events can be occurred in (m + n) ways.A permutation is an arrangement in a definite order of a number of objects, taken some or all at a time.Permutations when all the objects are distinct: The number of permutations of n different objects, taken r at a time, where r is less than zero and less than or equal to n and objects do not repeat is                       n (n – 1) (n – 2) … (n – r + 1), which is denoted by nPr or  P(n, r).n! = 1 × 2 × 3 × 4 × 5 × … × nThe number of permutations of n different objects taken r at a time, where repetition is not allowed, is given by nPr = n!/(n – r)!, where r is greater than or equal to zero and less than or equal to n.The number of permutations of n objects taken all at a time, where p1 objects are of first kind, p2 objects are of the second kind, …, pk objects are of kth kind and rest, if any, are all different is n!/(p1!p2!...pk!).The number of permutations of n different objects taken r at a time, where repetition is allowed, is nr.

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• Q1

A round table conference is to be held between 7 delegates of two countries. Find the number of ways in which they can be seated if 2 particular delegates are never to sit together.

Marks:2

Since the number of circular permutation of n different things = (n-1)!, therefore, the number of ways in which 7 delegates can be seated on round table = (7 – 1)! =6! Ways.

=6×5×4×3×2×1

=720

If 2 particular delegates like to seat together, then the number of seating arrangement = (6-1)! ×2!

= 5!×2

=5×4×3×2×1×2

=240

The number of ways in which they can be seated if 2 particular delegates are never to sit together=720 – 240

=480

• Q2

Evaluate 5P5 .

Marks:1

• Q3

Evaluate 5P3.

Marks:1

• Q4

How many permutations of the letters of the word ‘APPLE’ are possible?

Marks:1

Total letters = 5

A = 1 letter

P = 2 letters

L = 1 letter

E = 1 letter

Permutation =