Question

If all the words (with or without meaning) having five letters, formed using the letters of the word SMALL, and arranged as in a dictionary, then the position of the word SMALL is:

• A. ${46}^{th}$
• B. ${59}^{th}$
• C. ${52}^{nd}$
• D. ${58}^{th}$

Answer 1

Answer: (D)

${58}^{th}$

Position of SMALL in a dictionary :

Let us fix A in the first position.

Then, A _ _ _ _

At the $4$ places we want to arrange L,L,M and S which can be done in $\frac{4!}{2!}=12$ ways. (since 2 Ls are there)

Let us fix L in the first position.

Then, L _ _ _ _

At the $4$ places we want to arrange L,A,M and S which can be done in $4!=24$ ways.

Let us fix M in the first position.

Then, M _ _ _ _

At the $4$ places we want to arrange L,L,A and S which can be done in $\frac{4!}{2!}=12$ ways. (since 2 Ls are there)

Let us fix S in the first position and A in second position.

Then, S A _ _ _

At the $3$ places we want to arrange L,L and M which can be done in $\frac{3!}{2!}=3$ ways.

Let us fix S in the first position and L in second position.

Then, S L _ _ _

At the $3$ places we want to arrange L,A and M which can be done in $3!=6$ ways.

Let us fix M in second position.

Then, S M _ _ _

There arranging in alphabetical order, the next word is

S M A L L

Position $=12+24+12+3+6+1={58}^{th}$