Permutation and combination important formulas, tricks and shortcuts

Permuation formulas and shortcuts:

Factorial:Let n be a positive integer.Then factorial of n denoted by n! is defined as

n!=1*2*3*……………..(n-2)*(n-1)*n

Note:0!=1

Permutation: Different arrangement of a given number of things by taking some or all at a time.

Example:All arrangements made with letters a,b,c by taking two at a time are ab,ba,bc,cb,ca,ac.

All arrangements made with letters a,b,c by taking all at a time are abc,bca,cab,cba,acb,bac.

Number of permutation:Total number of possible arrangements(permutation) of n things, taken r at a time, is given by:

nPr =n!/(n-r)!=n(n-1)(n-2)………….(n-r+1)

Example 1.Arrangement of 3 items taken 2 at atime

3P2=3!/(3-1)!=3*2=6

Example2. Arrangement of 4 items taken all at a time

4P4=4!/(4-4)!=4!/0!=4*3*2*1=24

Permutation :Important point to note

Number of all permutation of n things all at a time is n!

If there are n objects ,m numbers are alike Then number of permutations of these objects is :n!/(m!)

If there are n objects ,p1 numbers are alike of one kind,p2 objects are alike of another kind,p3 are alike of third kind and so on and pr are alike of rth kind,such that p1+p2+p3+……pr=n Then number of permutations of these objects is : n!/(p1!.p2!…..pr!)

Combination shortcuts and formulas

Each of different groups or selection which can be formed by taking some or all of a number of object, is called a combination.

Suppose we want to select two students from a group of three students namely A,B and C.Then, possible selections are AB,BC and CA.

Note AB and BA represents same selection. But in permutation/arrangement AB and BA represents two different arrangements.

If we want to select ‘all at a time ‘, then there is only one possibility ABC.

Number of combinations:The number of all combination of n things, taken r at a time is:

nCr=n!/((r!)(n-r)!)=[n(n-1)(n-2)….upto r factors]/r!