Question

the number of ways in which we can select 5 letters of the word INTERNATIONAL is equal to?​

Answer 1

SOLUTION: 256 ways

ANSWER(EXPLAINATION):

We have thirteen letters in the word INTERNATIONAL

There are:

2 I's; 3 N's; 2T's; 2 A's and 1 each of E,O,L,R

There are 8 Types of Letters.

Then, We can select 5 letters like this:

By using COMBINATORICS, PERMUTATIONS, AND COMBINATIONS:

CASE 1:

=> All different

8 C 5​ = 8 P 5 / 5! ​=  [8!/ (8-5)!]/5! = (40320/6)/120 = 56 ways

(as, nCm = n P m/ m! , n P m = n!/(n-m)! )

Similarly,

CASE 2:

=> 2 alike, 3 different

 4C 1​ * 7 C 3​ = (4 P 1/ 1!) * (7 P 3/3!)  

 =  [4!/3!]/1! * [7!/4!]/3!

 = 24/6/1 * 5040/24/6

 = 4*35 = 140 ways

(as, we have 4 sets of alike letters)

CASE 3:

=> 3 alike, 2 different

1 C 1 * 7 C 2​ = (1 P 1 /1!) * (7 P 2/2!)

 = (1!/1!/1!) * (7!/5!)/2!

 = 1 * 5040/120/2

 = 1 * 21  = 21 ways

(we have only one set of 3 alike)

CASE 4:

=> 3 alike and 2 alike

1 C 1​ * 3 C 1 = (1 P 1/1!) * (3 P 1/1!)  

= (1!/1!/1!) * (3!/2!)/1!

= 1 * 6/2/1

= 1 * 3 = 3 ways

CASE 5:

=> Two sets of alike and one different

4 C 2​ * 6 C 1​ = (4 P 2/2!)*(6 P 1/1!)

=  (4!/2!)/2! * (6!/5!)/1!

= (24/2/2) * (720/120/1)

= 6 * 6 = 36 ways

THEREFORE,

TOTAL NUMBER OF COMBINATIONS: 56 + 140 + 21 + 3 + 36 = 256 ways