HOW TO FIND THE RANK OF A WORD IN DICTIONARY WITHOUT REPETITION

(i) GARDEN

Let us write the given word with alphabetical order,

A, D, E, G, N, R

Number of words starting with the letter A by rearranging the letters of the given word "GARDEN"

A _ _ _ _ _  =  5!

Number of words starting with the letter D

D _ _ _ _ _  =  5!

Number of words starting with the letter E

E _ _ _ _ _  =  5!

Number of words starting with the letter GAD

GAD _ _ _  =  3!

Number of words starting with the letter GAE

GAE _ _ _  =  3!

Number of words starting with the letter GAN

GAN _ _ _  =  3!

Number of words starting with the letter GAR

GAR _ _ _  =  1

The remaining letters are in alphabetical order.

3(5!) + 3(3!) + 1  =  3(120) + 3(6) + 1

  =  360 + 18 + 1

  =  379

Hence the total number of ways are 379.

(ii) DANGER

Let us write the given word with alphabetical order,

A, D, E, G, N, R

Number of words starting with the letter A by rearranging the letters of the given word "GARDEN"

A _ _ _ _ _  =  5!

Number of words starting with the letter DAE

D A E _ _ _  =  3!

Number of words starting with the letter DAG

D A G _ _ _  =  3!

Number of words starting with the letter DANE

D A N E _ _  =  2!

Number of words starting with the letter DANG

D A N G _ _  =  1

If we use the remaining letters in the alphabetical order.

5! + 2(3!) + 2! + 1  =  120 + 2(6) + 2 + 1

  =  120 + 12 + 3

  =  135

Hence the total number of ways are 135.

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