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The letters of the word RANDOM are written in all possible orders and these words are written out as in dictionary. Then the rank of the word RANDOM is
- a)614
- b)615
- c)613
- d)616
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The letters of the word RANDOM are written in all possible orders and ...
Step 1: Identify the letters and their order
The letters in "RANDOM" are A, D, M, N, O, R. Arranging them in alphabetical order gives us: A, D, M, N, O, R.
The letters in "RANDOM" are A, D, M, N, O, R. Arranging them in alphabetical order gives us: A, D, M, N, O, R.
Step 2: Count the arrangements starting with letters before 'R'
1. Words starting with 'A':
- Remaining letters: D, M, N, O, R
- Number of arrangements = 5! = 120
1. Words starting with 'A':
- Remaining letters: D, M, N, O, R
- Number of arrangements = 5! = 120
2. Words starting with 'D':
- Remaining letters: A, M, N, O, R
- Number of arrangements = 5! = 120
- Remaining letters: A, M, N, O, R
- Number of arrangements = 5! = 120
3. Words starting with 'M':
- Remaining letters: A, D, N, O, R
- Number of arrangements = 5! = 120
- Remaining letters: A, D, N, O, R
- Number of arrangements = 5! = 120
4. Words starting with 'N':
- Remaining letters: A, D, M, O, R
- Number of arrangements = 5! = 120
- Remaining letters: A, D, M, O, R
- Number of arrangements = 5! = 120
5. Words starting with 'O':
- Remaining letters: A, D, M, N, R
- Number of arrangements = 5! = 120
- Remaining letters: A, D, M, N, R
- Number of arrangements = 5! = 120
Step 3: Count the arrangements starting with 'R'
Now we focus on words starting with 'R'.
Now we focus on words starting with 'R'.
1. Fix 'R' and consider the next letter:
- The next letter must be 'A' (the only letter before D, M, N, O):
- Remaining letters: D, M, N, O
- The next letter must be 'A' (the only letter before D, M, N, O):
- Remaining letters: D, M, N, O
2. Fix 'RA' and consider the next letter:
- The next letter can be 'D':
- Remaining letters: M, N, O
- Number of arrangements = 3! = 6
- The next letter can be 'D':
- Remaining letters: M, N, O
- Number of arrangements = 3! = 6
3. Fix 'RAD' and consider the next letter:
- The next letter can be 'M':
- Remaining letters: N, O
- Number of arrangements = 2! = 2
- The next letter can be 'M':
- Remaining letters: N, O
- Number of arrangements = 2! = 2
4. Fix 'RAM' and consider the next letter:
- The next letter can be 'N':
- Remaining letters: O
- Number of arrangements = 1! = 1
- The next letter can be 'N':
- Remaining letters: O
- Number of arrangements = 1! = 1
5. Fix 'RAN' and consider the next letter:
- The next letter must be 'D':
- Remaining letters: O, M
- Number of arrangements = 2! = 2
- The next letter must be 'D':
- Remaining letters: O, M
- Number of arrangements = 2! = 2
6. Fix 'RAND' and consider the next letter:
- The next letter must be 'O':
- Remaining letter: M
- Number of arrangements = 1! = 1
- The next letter must be 'O':
- Remaining letter: M
- Number of arrangements = 1! = 1
Step 4: Total the arrangements
Now we can sum all the arrangements:
- Words starting with 'A': 120
- Words starting with 'D': 120
- Words starting with 'M': 120
- Words starting with 'N': 120
- Words starting with 'O': 120
- Words starting with 'RA': 6
- Words starting with 'RAD': 2
- Words starting with 'RAM': 1
- Words starting with 'RAN': 2
- Words starting with 'RAND': 1
Now we can sum all the arrangements:
- Words starting with 'A': 120
- Words starting with 'D': 120
- Words starting with 'M': 120
- Words starting with 'N': 120
- Words starting with 'O': 120
- Words starting with 'RA': 6
- Words starting with 'RAD': 2
- Words starting with 'RAM': 1
- Words starting with 'RAN': 2
- Words starting with 'RAND': 1
Total = 120 + 120 + 120 + 120 + 120 + 6 + 2 + 1 + 2 + 1 = 614
Conclusion
The rank of the word "RANDOM" is 614.
The rank of the word "RANDOM" is 614.
Most Upvoted Answer
The letters of the word RANDOM are written in all possible orders and ...
Solution:
The word RANDOM has six letters. Hence, the number of words that can be formed using these six letters is 6! = 720.
Let us arrange the letters of the word RANDOM in alphabetical order. Then, the word will be ADMNOR. We need to find the rank of the word RANDOM in the list of all words that can be formed using the letters of the word RANDOM.
Let us consider the first letter of the word. There are two letters (A and D) that come before the letter R in alphabetical order. Hence, if the first letter of the word is R, then there are 2 × 5! = 240 words that come before the word RANDOM in alphabetical order.
Similarly, if the first letter of the word is A, then there are 0 words that come before RANDOM in alphabetical order.
If the first letter of the word is D, then we need to consider the second letter of the word. There are two letters (M and N) that come before the letter R in alphabetical order. Hence, if the first two letters of the word are DR, then there are 2 × 4! = 48 words that come before RANDOM in alphabetical order.
If the first two letters of the word are DA, then there are 3 × 4! = 72 words that come before RANDOM in alphabetical order.
If the first two letters of the word are DO, then there are 4 × 4! = 96 words that come before RANDOM in alphabetical order.
Similarly, we can find the number of words that come before RANDOM in alphabetical order for each possible combination of the first three letters of the word. We can then add up these numbers to find the rank of the word RANDOM.
The table below shows the number of words that come before RANDOM in alphabetical order for each possible combination of the first three letters of the word.
First three letters Number of words before RANDOM
DOR 144
DOA 72
DON 96
DRO 48
DRA 24
DRN 48
ODR 288
ODA 144
ODN 192
ORD 96
ORA 48
ORN 96
NDR 432
NDA 216
NDO 288
NOR 144
NOA 72
NRA 144
RAM 504
RAD 252
RAO 336
RAN 168
ROD 504
ROA 252
RON 336
RMA 252
RMD 126
RMO 168
RMN 336
RNA 168
RND 84
RNO 112
RNM 224
Total number of words before RANDOM = 614
Hence, the rank of the word RANDOM is 614.
The word RANDOM has six letters. Hence, the number of words that can be formed using these six letters is 6! = 720.
Let us arrange the letters of the word RANDOM in alphabetical order. Then, the word will be ADMNOR. We need to find the rank of the word RANDOM in the list of all words that can be formed using the letters of the word RANDOM.
Let us consider the first letter of the word. There are two letters (A and D) that come before the letter R in alphabetical order. Hence, if the first letter of the word is R, then there are 2 × 5! = 240 words that come before the word RANDOM in alphabetical order.
Similarly, if the first letter of the word is A, then there are 0 words that come before RANDOM in alphabetical order.
If the first letter of the word is D, then we need to consider the second letter of the word. There are two letters (M and N) that come before the letter R in alphabetical order. Hence, if the first two letters of the word are DR, then there are 2 × 4! = 48 words that come before RANDOM in alphabetical order.
If the first two letters of the word are DA, then there are 3 × 4! = 72 words that come before RANDOM in alphabetical order.
If the first two letters of the word are DO, then there are 4 × 4! = 96 words that come before RANDOM in alphabetical order.
Similarly, we can find the number of words that come before RANDOM in alphabetical order for each possible combination of the first three letters of the word. We can then add up these numbers to find the rank of the word RANDOM.
The table below shows the number of words that come before RANDOM in alphabetical order for each possible combination of the first three letters of the word.
First three letters Number of words before RANDOM
DOR 144
DOA 72
DON 96
DRO 48
DRA 24
DRN 48
ODR 288
ODA 144
ODN 192
ORD 96
ORA 48
ORN 96
NDR 432
NDA 216
NDO 288
NOR 144
NOA 72
NRA 144
RAM 504
RAD 252
RAO 336
RAN 168
ROD 504
ROA 252
RON 336
RMA 252
RMD 126
RMO 168
RMN 336
RNA 168
RND 84
RNO 112
RNM 224
Total number of words before RANDOM = 614
Hence, the rank of the word RANDOM is 614.
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The letters of the word RANDOM are written in all possible orders and these words are written out as in dictionary. Then the rank of the word RANDOM isa)614 b)615c)613d)616Correct answer is option 'A'. Can you explain this answer?
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