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Q.
Number of words that can be formed using all the letters of the word GARGEE if no two alike letters are together, is
Permutations and Combinations
Solution:
Total $- n ( A \cup B )=\frac{6 !}{2 ! 2 !}-( n ( A )+ n ( B )- n ( A \cap B ))$
$=\frac{6 !}{2 ! 2 !}-\left(\frac{5 !}{2 !}+\frac{5 !}{2 !}-4 !\right)=180-96=84$
Set $A$ represents number of ways when $G's$ are together
Set B represents number of ways then E's are together
Aliter: GGEEAR
Number of words when G's are separated
$=\frac{4 !}{2 !} \cdot{ }^{5} C _{2}=120$
Number of words when G's are separated but E's are together $=3 ! \times{ }^{4} C_{2}=36$
$\therefore $ Number of ways when no two alike letters are together
$=120-36=84$
