Question

If the letters of the word PRISON are permuted in all possible ways and the words thus formed are arranged in dictionary order, find the rank of the word PRISON.

Answer 1

Total no. of letters in the word PRISON $=6$

If all permutations are arranged alphabetically, then all words starting with $I,N,O,PI,PN,PO,PRIN,PRIO$ and $PRISN$ will be before PRISON.

No. of words starting with $I=5!=120$

Similarly,

No. of words starting with $N=5!=120$

No. of words starting with $O=5!=120$

No. of words starting with $PI=4!=24$

No. of words starting with $PN=4!=24$

No. of words starting with $PO=4!=24$

No. of words starting with $PRIN=2!=2$

No. of words starting with $PRIO=2!=2$

No. of words starting with $PRISN=1!=1$

Total no. of words before word PRISON $=120+120+120+24+24+24+2+2+1=437$
Since, the rank of a given word is basically finding out the position of the word when all possible words have been formed.

Thus, the ${438}^{th}$ word will be PRISON.

Hence, the rank of the word PRISON is $438$.