# Permutations and Combinations-algebra tutors

### Permutations and Combinations(part-2)

In my previous post, we discussed the fundamental principle of counting and various methods of permutations. In this post, I shall discuss combinations in details.

Meaning of Combination- If we are given a set of objects and we want to select a few objects out of this set, then we can do it by many different ways. These ways are known as combinations.
Example- If we are given three balls marked as B, W and R and we want to select two balls then we can select like this- BW, BR, WR.

These are known as the combination of this selection.

Combination of n different objects taken r at a time when repetition is not allowed– If repetition is not allowed the number of ways of selecting r objects out of a group of n objects is called

=

In latest notation system   is also known as C(n;r) or

Properties of  – It’s a very useful and interesting Mathematical tool. It has following properties.

(i)

(ii)

(iii)      known as Pascal’s law

(iv) r.

(v)

(vi) If n is even then we should put r=n/2 for maximum value of   and if n is odd then   is greatest when r=

(vii) In the expansions of  if we put x=1 then

Some difficult Combinations-

(a) If we want to select r objects out of a set of n objects such that p particular objects are always selected then total number of ways =

(b) If we want to select r objects out of a set of n objects such that p particular objects are never selected then total number of ways =

(c) If we want to select one or more object(at least one object) out of a set of n objects then total number of ways=

(d) If we want to select zero or more object(at least zero object) out of a set of n objects then total number of ways=

(e) If we want to select two or more object(at least two objects) out of a set of n objects then total number of ways=

(f) If we want to select one or more object(at least one object) out of a set of n identical objects then total number of ways=  = n

(g) If we have a set of objects in which p objects are of one type, q objects are of other type and r objects are of some other type and we want to select some or all out of this set then total number of combinations=(p+1)(q+1)(r+1)-1

(h) If we have a set of objects in which p objects are of one type, q objects are of other type, r objects are of some other type, n objects are different and we want to select at least one out of this set then total number of combinations=(p+1)(q+1)(r+1)

(i) If we have a set of objects in which p objects are of one type, q objects are of other type, r objects are of other type, n objects are different and we want to select such that at least one object of each kind is included total number of combinations= pqr( )

(j) Number of ways in which (m+n+p) can be divided into three unequal groups=

(h) Number of ways in which (m+n+p) can be divided and distributed into three unequal groups=

Problems where we use both permutations and combinations- In exams, usually we are asked questions in which we have to use both permutations and combinations.

(a) The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always included= . r!

(b) The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always excluded =  .r!

(c) The number of ways of selecting and arranging r objects out of a group of n different objects such that p particular objects are always seperated=

De-arrangement theorem- This theorem is used to permutate n distinct objects such that no object gets its original position.

according to this theorem, total no. of ways=

Formation of Geometrical Figures Using Combinations-

(a) If we are given n points on a plane then, the number of lines formed by joining these points =

(b) If we are given n points on a plane out of which m are in a straight line, the number of lines formed by joining these points=

(c) If we are given n points on a plane then, the number of triangles formed by joining these points =

(d) If we are given n points on a plane out of which m are in a straight line, the number of triangles formed by joining these points=

(e) If we are given n points on a plane, the number of diagonals formed by joining these points=  =

Here I am posting a pdf. Use concepts that are given in this post and previous post to solve questions given in the pdf