Question

If the different permutations of the word PRODIGIOUS are listed as in a dictionary, then what is the rank of the word PRODIGIOUS?

• A. 200966
• B. 609902
• C. 12500
• D. 24800

Answer 1

The correct option is B

609902

Correct order of the letters: D, G, I, I, O, O, P, R, S, U
$\begin{array}{cc}\text{Word begin with}& \text{Number of words}\\ D& \frac{9!}{2!2!}\\ G& \frac{9!}{2!2!}\\ I& \frac{9!}{2!}\\ O& \frac{9!}{2!}\\ PD& \frac{8!}{2!2!}\\ PG& \frac{8!}{2!2!}\\ PI& \frac{8!}{2!}\\ PO& \frac{8!}{2!}\\ PRD& \frac{7!}{2!2!}\\ PRG& \frac{7!}{2!2!}\\ PRI& \frac{7!}{2!}\\ PRODG& \frac{5!}{2!}\end{array}$
Now the next two words are PRODIGIOSU and PRODIGIOUS
Hence the rank of the word PROFIGIOUS
$=2\left(\frac{9!}{2!2!}\right)+2\left(\frac{9!}{2!}\right)+2\left(\frac{8!}{2!2!}\right)+2\left(\frac{8!}{2!}\right)+2\left(\frac{7!}{2!2!}\right)+\frac{7!}{2!}+\frac{5!}{2!}+2=3\left(\frac{9!}{2!}\right)+3\left(\frac{8!}{2!}\right)+7!+\left(\frac{5!}{2!}\right)+2=544320+60480+5040+60+2=609902$