Question

Find the position of the word SMALL, if all the words having five letters formed using the letters of the word SMALL and arranged as in the dictionary.
$\begin{align}
  & \left( A \right){{58}^{th}} \\
 & \left( B \right){{46}^{th}} \\
 & \left( C \right){{59}^{th}} \\
 & \left( D \right){{42}^{th}} \\
\end{align}$

Answer 1

Hint: We solve the question by arranging the letters one by one in alphabetic order and counting the number of possible words until we get our required word SMALL. We use the formula, the number of the arrangement of N letters is $N!$ and the number of the arrangement of N letters in which a letter is repeated n times is $\dfrac{N!}{n!}$ Then we need to add one as the count we got is the number of words that occurred before our required word.

Complete step-by-step solution:
We were given the word SMALL.
Now, arranging all the letters in alphabetical order, we have
A, L, L, M, S.
Now, let us consider all the possible words that can be formed using the given letters and starting with A.
\[\begin{matrix}
   A & \underline{{}} & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
Fixing A in the first position, we need to arrange the letters L, L, M, S.
Using the formula for the number of arrangements of N objects out of which one object is repeated n times,
$\Rightarrow \dfrac{N!}{n!}$
The possible number of words is equal to the number of ways of arranging the remaining letters. Here the total number of objects to arrange are 4 and L is repeated 2 times. So, using the above formula we get,
$\Rightarrow \dfrac{4!}{2!}=12..........\left( 1 \right)$
So, the number of words starting with A is 12.
Fixing L in the first position, we need to arrange the letters A, L, M, S.
\[\begin{matrix}
   L & \underline{{}} & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
Let us consider the formula for number of arrangements of N objects,
$\Rightarrow N!$
The possible number of words is equal to number of ways of arranging the remaining letters. Using the formula for arrangement we get,
$\Rightarrow 4!=24..........\left( 2 \right)$
So, the number of words starting with L are 24.
Now, let us consider all the possible words that can be formed using the given letters and starting with M.
\[\begin{matrix}
   M & \underline{{}} & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
Fixing M in the first position, we need to arrange the letters A, L, L, S.
The possible number of words is equal to the number of ways of arranging the remaining letters. Using the formula for arrangement we get,
$\Rightarrow \dfrac{4!}{2!}=12..........\left( 3 \right)$
So, the number of words starting with M is 12.
Now, let us consider all the possible words that can be formed using the given letters and starting with S.
\[\begin{matrix}
   S & \underline{{}} & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
As we need the word SMALL, we fix S and place the other letters.
Now, let us consider all the possible words that can be formed using the given letters and starting with A.
\[\begin{matrix}
   S & A & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
Now, we need to arrange the letters L, L, M.
The possible number of words is equal to the number of ways of arranging the remaining letters. Using the formula for arrangement we get,
$\Rightarrow \dfrac{3!}{2!}=3..........\left( 4 \right)$
So, the number of words starting with S, A is 3.
Now, let us consider all the possible words that can be formed using the given letters and starting with L.
\[\begin{matrix}
   S & L & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
Now, we need to arrange the letters A, L, M.
The possible number of words is equal to the number of ways of arranging the remaining letters. Using the formula for arrangement we get,
$\Rightarrow 3!=6..........\left( 5 \right)$
So, the number of words starting with S, L is 6.
Now, let us consider all the possible words that can be formed using the given letters and starting with M.
\[\begin{matrix}
   S & M & \underline{{}} & \underline{{}} & \underline{{}} \\
\end{matrix}\]
As we need the letters starting with S, M fix M and arrange the letters A, L, L.
Now, let us consider all the possible words that can be formed using the given letters and starting with A.
\[\begin{matrix}
   S & M & A & \underline{{}} & \underline{{}} \\
\end{matrix}\]
As we need the letters starting with S, M, A fix A and arrange the letters L, L.
Now, let us consider all the possible words that can be formed using the given letters and starting with L.
\[\begin{matrix}
   S & M & A & L & \underline{{}} \\
\end{matrix}\]
As we need the letters starting with S, M, A, L fix L and arrange the letter L.
\[\begin{matrix}
   S & M & A & L & L \\
\end{matrix}\]
As it is the letter only left, we get the word SMALL.
So, the number of words occurred before the occurrence of the word SMALL are,
Using equations (1), (2), (3), (4), (5), we get
\[\begin{align}
  & =12+24+12+3+6 \\
 & =57 \\
\end{align}\]
So, the word SMALL is the ${{58}^{th}}$ word.
Hence, the answer is ${{58}^{th}}$. Hence the answer is Option A.

Note: As the letter L is repeated 2 times, one might consider the letters starting with L two times. But those words are the same taken with any of the two L. So, we need to consider the words starting with the letter L only once.