Aptitude Day 122

Problems on Permutations and Combinations:

1.In how many different ways can the letters of the word 'EAT' be arranged in such a way that the vowels occupy only the odd positions?

Solution:
               There are 3 letters in the given word.
               The positions are
                                             1     2     3
             
               Now, the 2 vowels E and A can be placed at 2 odd positions 1 and 3.
               The consonant T will occupy the even position 2.

               The number of ways of arranging the vowels = ²P2 = 2! = 2.
               The number of ways of arranging the consonant =   ¹P1 = 1! = 1.

               Total number of ways = 2*1 = 2.

2. In how many different ways can the letters of the word 'TABLE' be arranged in such a way that the vowels occupy only the odd positions?

Solution:
             There are 5 letters in the given word.
               The positions are
                                             1     2     3     4     5
           
               Now, the 2 vowels A and E can be placed at any of the 3 odd positions 1,3 and 5.
               The consonants will occupy the remaining positions.

               The number of ways of arranging the vowels = ³P2 = 3*2*1 = 6.
               The number of ways of arranging the consonant =   ³P3 = 3*2*1 = 6.

               Total number of ways = 6*6 = 36.