Aptitude Day 126

Problems on Permutations and Combinations:

1. In how many ways can 18 books on Mathematics and 15 books on English be placed in a row on a shelf so that two books on English may not be together?

Solution:
               The arrangement will be
                               E M E M E M ..... E M _ M _ M _ M_

              So 15 places out of 19 can be chosen =  ¹⁹C15
                                                                          =  ¹⁹C4
                                                                          = 19*18*17*16/ 1*2*3*4
                                                                          = 3876.


2. Out of 5 consonants and 3 vowels, how many words of 2 consonants and 1 vowel can be formed?

Solution:
               2 Consonants out of 5 and 1 vowel out of 3 =  ( ⁵C2 * ³C1) = 5*4/1*2 *3
              The Number of groups, each having  2 consonants and 1 vowel = 30.

              Each group contains 3 letters.
              The number of ways of arranging 3 letters among themselves = 3! = 6.

              The required number of words = 30*6 = 180.