Prove that there are only seven number of ways in which any four letters can be selected out of the letters of the word GORCOO.

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Solution

We have 6 letters out of which 3 are alike of one type and 3 different but in all 4 types: we can select them in the following manner:
(i) All different $= {^{C}}_{4} = 1$
(ii) 2 alike 2 different $= {^{1} C}_{1} {3 C}_{2} = 1.3 = 3$
(iii) 3 alike and 1 different $= {^{1} C}_{1} {3 C}_{1} = 1.3 = 3$
Hence the total number of ways is $1 + 3 + 3 = 7$
Out of there alike $O O O$ you may select one $O$ or two $O$ or three $O$ , this can be done only in one way.

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