Solved Examples

Example 1

A letter lock consists of three rings each marked with 10 different letters. In how many ways is it possible to make an unsuccessful attempt to open the block?

Solution:

Two rings may have same letter at a time but same ring cannot have two letters at  time, therefore, we must proceed ring wise.

   Each of the three rings can have any one of the 10 different letters in 10 ways.

   Therefore  Total number of attempts = 10 × 10 × 10 = 1000.

   But out of these 1000 attempts only one attempt is successful.

  Therefore Required number of unsuccessful attempts

   =      1000 - 1 = 999.

Example 2

Find the total number of signals that can be made by five flags of different colour when any number of them may be used in any signal.

Solution:

Case I :     When only one flag is used.

                No. of signals made = ⁵P₁ = 5.

Case II :    When only two flag is used.

                Number of signals made = ⁵P₂ = 5.4 = 20.

Case III :   When only three flags are used.

                Number of signals is made = ⁵P₃ = 5.4.3 = 60.

Case IV :   When only four flags are used.

                Number of signals made = ⁵P₄ = 5.4.3.2 = 120.

Case V :     When five flags are used.

                Number of signals made = ⁵P₅ = 5! = 120.

Hence, required number = 5 + 20 + 60 + 120 + 120 = 325.

Example 3

Prove that if each of the m points in one straight line be joined to each of the n points on the other straight line, the excluding the points on the given two lines. Number of points of intersection of these lines is 1/4 mn
(m-1(n-1).

Solution:

To get one point of intersection we need two points on the first line and two points on the second line. These can be selected out of n-points in ⁿC₂ ways and for m points in ^mC₂ ways.

       Therefore Required number = ^mC₂ × ⁿC₂

        = (m(m-1))/2! . (n(n-1))/2!

        = 1/4 m n (m - 1)(n - 1)

Example 4

There are ten points in a plane. Of these ten points four points are in a straight line and with the exception of these four points, no other three points are in the same straight line. Find

       (i) The number of straight lines formed.

       (ii) The number of triangles formed.

       (iii) The number of quadrilaterals formed by joining these ten points.