From the string   a b a c a b a b a b c d c e d , if 5 letters should be selected, then the number of ways in which this selection can be done is

Step-by-step solution

In the string “ a b a c a b a b a b c d c e d ”   there are five a ' s , four b ' s , three c ' s , two d ' s and one e Case(I): 5 Unique letters = 5 C 5 = 1 Case(ii): 2 Same and 3 Different = 4 C 1 × 4 C 3 = 16 Case(iii): 2 Same, 2 More Same and 1 Unique = 4 C 2 × 3 C 1 = 18 Case(iv): 3 same, 2 Unique = 4 C 2 × 3 C 1 = 18 case (v): 3 same, 2 same = 3 C 1 × 3 C 1 = 9 Case(vi): 4 same, 1 unique = 2 C 1 × 4 C 1 = 8 case(vii): all 5 same = 1 Hence, total number of ways = 71