Question

The letters of the word SACHIN are permuted and are arranged in an alphabetical order as in an English dictionary. Then, the rank of the word SACHIN is _____

___

Answer 1

Let us arrange the letters of the word SACHIN in the alphabetical order.

ACHINS

In the dictionary, the words appearing first will begin with A, then C, then H, then I, then N and at last S.

Let us fix the first position as A.

A _ _ _ _ _

The 5 places above can be filled with letters C H I N S in ${}^5{P}_5$ = 5! ways.

So,there are 5! i.e., 120 words possible with first letter being A

Similarly, fixing the first letter as C, the remaining 5 places can be filled with letters A H I N S in ${}^5{P}_5$ = 5! ways

That is,

C _ _ _ _ _ $\to$5! words possible

Proceeding in the similar way, number of words in the dictionary are

H _ _ _ _ _ $\to$5! words

I _ _ _ _ _ $\to$5! words

N _ _ _ _ _ $\to$5! words

Now, we are left with words with $1^{st}$ letter as S

The first word starting with S found in the dictionary is SACHIN as other letters A,C,H, I, N are in the alphabetical order.

So, before SACHIN, we calculated that there exists $5\times 5!$ words in the dictionary.

Therefore, the rank of the word SACHIN = 601