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Q.
The total number of ways of selecting five letters from the letters of the word 'INDEPENDENT' is
Rajasthan PETRajasthan PET 2010
Solution:
Word 'INDEPENDENT' here, $ 1-1,N-3,D $ $ 2,E-3,P-1,T-1 $
Selection can be made by the following number of ways
(i) If all the letters are distinct, then no. of ways
$ {{=}^{6}}{{C}_{5}}=6 $
(ii) If three are different and two letters are same, then no. of way $ {{=}^{3}}{{C}_{1}}{{.}^{5}}{{C}_{3}}=30 $
(iii) If two letters are different and three are same, then ways
$ {{=}^{2}}{{C}_{1}}{{.}^{5}}{{C}_{2}}=20 $
(iv) If two letters are same another two letters are same and one letter is different, then no. of ways
$ {{=}^{3}}{{C}_{2}}{{.}^{4}}{{C}_{1}}=12 $
(v) If three letters are same and another two letters are same, then ways
$ {{=}^{2}}{{C}_{1}}{{.}^{2}}{{C}_{1}}=4 $
$ \therefore $ Total number of ways
$ =6+30+20+12+4=72 $