$AGAIN$ शब्द के अक्षरों से बनने वाले, अर्थपूर्ण या अर्थहीन, शब्दों की संख्या ज्ञात कीजिए। यदि इन शब्दों को इस प्रकार लिखा जाए जिस प्रकार किसी शब्दकोश में लिखा जाता है, तो $50$ वाँ शब्द क्या है ?

There are $5$ letters in the word $AGAIN$, in which $A$ appears $2$ times. Therefore, the required number of words $=\frac{5 !}{2 !}=60$

To get the number of words starting with $A$, we fix the letter $A$ at the extreme left position, we then rearrange the remaining $4$ letters taken all at a time. There will be as many arrangements of these $4$ letters taken $4$ at a time as there are permutations of $4$ different things taken $4$ at a time. Hence, the number of words starting with $A=4 !=24 .$ Then, starting with $G$, the number of words $=\frac{4 !}{2 !}=12$ as after placing $G$ at the extreme left position, we are left with the letters $A , A , I$ and $N$. Similarly, there are $12$ words starting with the next letter $I$. Total number of words so far obtained $=24+12+12=48$

The $49^{\text {th }}$ word is $NAAGI$. The $50^{\text {th }}$ word is $NAAIG$.