Question

Find the total number of ways of selecting five letters from the letters of the word INDEPENDENT.

Answer 1

Given:  

• Letters of the word INDEPENDENT.

To Find:

• Total number of ways of selecting five letters from the letters of the word INDEPENDENT.

Solution:

Letters of the word INDEPENDENT are I , P , T , DDD , NNN , EEE

Now, Five letters can be selected in the following ways :

• When all letters are different then

                   Number of ways $=^6C_5$ $=6$

• When two letters are same and three are different then,

                   Number of ways$= ^3c_1 \times ^5 c_3$

                                              $=3 \times 10=30$

• When three letters are same and two are different then,

                    Number of ways = $^2c_1\times ^5c_2$

                                                 $=2\times 10=20$

• When three letters are same and two are same then,

                    Number of ways  =  $^2c_1 \times ^2c_1$

                                                  $= 2 \times 2=4$

• When two letters are same , two are same and one is different then,

                    Number of ways $= ^3c_2 \times ^4c_1$

                                                 $=3 \times 4=12$

        Total selections

                                $=6+30+20+4+12\\ =72$

Thus, Total number of ways of selecting five letters $= 72ways$