If the letters of the word $GARDEN$ are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
- $\; GARDEN$
- $\; DANGER$
The given word is $GARDEN$.
The dictionary order of the letters of given word is $\; A, \; D, \; E, \; G, \; N, \; R$
- In the dictionary order, words which begin with $A$ come first.
If the first place is filled with $A$, the remaining 5 letters $\; D, \; E, \; G, \; N, \; R \;$ can be arranged in $5!$ ways.
Proceeding in this manner we have,
$A \; - \; - \; - \; - \; - \; = 5!$ ways
$D \; - \; - \; - \; - \; - \; = 5!$ ways
$E \; - \; - \; - \; - \; - \; = 5!$ ways
$G \; A \; D \; - \; - \; - \; = 3!$ ways
$G \; A \; E \; - \; - \; - \; = 3!$ ways
$G \; A \; N \; - \; - \; - \; = 3!$ ways
$G \; A \; R \; D \; E \; N \; = 1$ way
$\therefore \;$ Rank of the word $GARDEN$ is $= 3 \times 5! + 3 \times 3! + 1 = 3 \times 120 + 3 \times 6 + 1 = 379$ - Rank of the word $DANGER$
$A \; - \; - \; - \; - \; - \; = 5!$ ways
$D \; A \; E \; - \; - \; - \; = 3!$ ways
$D \; A \; G \; - \; - \; - \; = 3!$ ways
$D \; A \; N \; E \; - \; - \; = 2!$ ways
$D \; A \; N \; G \; E \; R \; = 1$ way
$\therefore \;$ Rank of the word $DANGER$ is $= 5! + 2 \times 3! + 2! + 1 = 120 + 12 + 2 + 1 = 135$