Question

How many ways can you arrange the letters of the word “COURSE” if: The first and last letters must be a vowel.

Answer 1

The unique alphabets in Course are : C , E , O , R , S , U

No . of vowels = 3 (E,O,U)

No. of Consonants = 3 ( C,R,S)

We use Fundamental Principle of counting which states: if there are n1 ways to do first action , n2 ways to do second action and so on . Then no of ways to do 'n' actions together = n1 * n2 * n3 ------------ * nn   

We think of each space for a letter as a place in which we have to fill a letter

So we can consider filling of each place in the 6 letter word as separate actions.

So Total no. of ways to arrange words of Course such that 1st and last letters are vowel =

=no. of ways to fill first place * no. of ways to fill second place * -----* no. of ways to fill 6th place

No. of ways to select 'r' objects out of 'n' different objects =

No. of ways to fill first place with a vowel = we select a vowel out of 3 vowels used in Course = = 3

No. of ways to fill last place with a vowel = we select a vowel out of 2 remaining vowels = = 2

No. of ways to fill second place = we can select any one of the remaining 4 letters = = 4

No. of ways to fill third place = we can select any one of the remaining 3 letters = =3

Similarly ,no .of ways to fill fourth and fifth place are and respectively .

So no. of ways to fill each place is as given below:

Total no. of ways to arrange letters of word 'COURSE' such that 1st and last letters are vowels = 3*4*3*2*1    *2

   = 144

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