Question

The letter of the word RANDOM are written in all possible orders and these words are written out as dictionary then the rank of the word RANDOM is

Answer 1

Here we will arrange the given word in dictionary order i.e. first we will calculate the number of words starting with A, D, M, N, O and then we will find the rank of the word RANDOM int the words starting with R.

Complete step-by-step solution:
We have to arrange the word RANDOM in dictionary order.
Hence first we need to find the number of all the possible words starting with A.
Hence let us fix ‘A’ at the first place:

A| | | | |
---|---|---|---|---|---

Then, the second place can be filled by 5 ways.
Third place can then be filled by 4 ways.
Fourth place can be filled in 3 ways.
Fifth place can be filled in 2 ways.
And, the sixth place can be filled by 1 way.
Hence the total number of words starting with A are $5 \times 4 \times 3 \times 2 \times 1$ words$= 120$ words
Now we will find the number of words starting with D as the next letter of the given word in the dictionary is D.
Hence let us fix ‘D’ at the first place:

D| | | | |
---|---|---|---|---|---

Then, the second place can be filled by 5 ways.
Third place can then be filled by 4 ways.
Fourth place can be filled in 3 ways.
Fifth place can be filled in 2 ways.
And, the sixth place can be filled by 1 way.
Hence the total number of words starting with D are $5 \times 4 \times 3 \times 2 \times 1$words $= 120$ words
Now we will find the number of words starting with M as the next letter of the given word in the dictionary is M.
Hence let us fix ‘M’ at the first place:

M| | | | |
---|---|---|---|---|---

Then, the second place can be filled by 5 ways.
Third place can then be filled by 4 ways.
Fourth place can be filled in 3 ways.
Fifth place can be filled in 2 ways.
And, the sixth place can be filled by 1 way.
Hence the total number of words starting with M are $5 \times 4 \times 3 \times 2 \times 1$words $= 120$ words
Now we will find the number of words starting with N as the next letter of the given word in the dictionary is N.
Hence let us fix ‘N’ at the first place:

N| | | | |
---|---|---|---|---|---

Then, the second place can be filled by 5 ways.
Third place can then be filled by 4 ways.
Fourth place can be filled in 3 ways.
Fifth place can be filled in 2 ways.
And, the sixth place can be filled by 1 way.
Hence the total number of words starting with N are $5 \times 4 \times 3 \times 2 \times 1$words $= 120$ words
Now we will find the number of words starting with O as the next letter of the given word in the dictionary is O.
Hence let us fix ‘O’ at the first place:

O| | | | |
---|---|---|---|---|---

Then, the second place can be filled by 5 ways.
Third place can then be filled by 4 ways.
Fourth place can be filled in 3 ways.
Fifth place can be filled in 2 ways.
And, the sixth place can be filled by 1 way.
Hence the total number of words starting with O are $5 \times 4 \times 3 \times 2 \times 1$words $= 120$words
Now we will calculate the number of words starting with R.
The first word starting with R in a dictionary would be RADMNO.
Now fixing the first three positions we will calculate the number of words starting with RAD.

R| A| D| | |
---|---|---|---|---|---

Now the fourth place can be filled by 3 ways
Fifth place can be filled by 2 ways
Sixth place can be filled by 1way.
Hence the number of words starting with RAD are $3 \times 2 \times 1$
$= 6$words
Now the next word in the dictionary would be RAMDNO
Now fixing the first three positions we will calculate the number of words starting with RAM.

R| A| M| | |
---|---|---|---|---|---

Now the fourth place can be filled by 3 ways
Fifth place can be filled by 2 ways
Sixth place can be filled by 1way.
Hence the number of words starting with RAM are $3 \times 2 \times 1$
$= 6$words
The next two words will be
RANDMO and RANDOM.
Hence the rank of RANDOM is :-
$= 120 + 120 + 120 + 120 + 120 + 6 + 6 + 2$
$= 614$

Hence the rank of RANDOM is 614.

Note: In such questions, we have to arrange all the letters of the given word in alphabetical order as the words in a dictionary are arranged in alphabetical order and then calculate the number of words to get the rank of a particular word.
Students should note that for arrangement of elements we use permutation and for choosing elements we use the formula for combinations.