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$AGAIN$ શબ્દના બધા મૂળાક્ષરોનો ઉપયોગ કરીને અર્થસભર કે અર્થરહિત કેટલા શબ્દો બનાવી શકાય તે શોધો. જો આ શબ્દોને શબ્દકોષ પ્રમાણે લખ્યા હોય, તો $50$ મા સ્થાને કયો શબ્દ આવે ?
$NAAIG$
$NAAIG$
$NAAIG$
$NAAIG$
Solution
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 $NAAIG$.