Six letter words are formed using all the letters of the word NUMBER. If these words are arranged in the dictionary order, the rank of the word NUMBER is 

We are tasked with finding the rank of the word NUMBER when all possible 6-letter words are formed using the letters of the word "NUMBER" and arranged in lexicographical (dictionary) order. Let's break down the process step by step.

Step 1: List the letters in alphabetical order

The word "NUMBER" consists of the following letters: N, U, M, B, E, R. When arranged alphabetically, the order of letters is:

• B, E, M, N, R, U

Step 2: Calculate how many words precede "NUMBER"

Now, let's calculate how many 6-letter words precede the word "NUMBER" in lexicographical order.

Starting with 'N'

The first letter of the word "NUMBER" is 'N'. Let's count how many words start with letters that come before 'N' (i.e., 'B', 'E', 'M').

• For each letter before 'N' (i.e., 'B', 'E', 'M'), the remaining 5 letters can be rearranged in 5! ways. Thus, for each of 'B', 'E', and 'M', there are:

5! = 120 words

• So, there are 3 * 120 = 360 words that start with 'B', 'E', or 'M'.

Now, consider words starting with 'N'

Next, we focus on words starting with 'N'. The second letter in "NUMBER" is 'U'. Let's count the words that begin with 'N' and the second letter before 'U' (i.e., 'B', 'E', 'M').

• For each letter before 'U' (i.e., 'B', 'E', 'M'), the remaining 4 letters can be rearranged in 4! ways. So, for each of 'B', 'E', and 'M', there are:

4! = 24 words

• Thus, for letters 'B', 'E', and 'M', there are 3 * 24 = 72 words that start with 'N' and a letter before 'U'.

Now, consider words starting with 'NU'

We now focus on words starting with "NU". The third letter in "NUMBER" is 'M'. Let's count how many words start with 'NU' and a third letter before 'M' (i.e., 'B', 'E').

• For each of 'B' and 'E', the remaining 3 letters can be rearranged in 3! ways. So, for 'B' and 'E', there are:

3! = 6 words

• Thus, for 'B' and 'E', there are 2 * 6 = 12 words that start with 'NU' and a letter before 'M'.

Now, consider words starting with 'NUM'

We now focus on words starting with "NUM". The fourth letter in "NUMBER" is 'B'. Let's count how many words start with 'NUM' and a fourth letter before 'B' (i.e., 'E').

• For 'E', the remaining 2 letters can be rearranged in 2! ways. So, for 'E', there are:

2! = 2 words

• Thus, there are 2 words that start with 'NUM' and a letter before 'B'.

Finally, consider the word "NUMBER"

Now, we are left with "NUMBER" itself. As it is the exact match, the count for this word is 1.

Step 3: Calculate the Rank

The rank of "NUMBER" can be calculated by adding all the preceding words and including "NUMBER" itself. The total number of words preceding "NUMBER" is:

• 360 (from 'B', 'E', 'M')
• + 72 (from 'NU' and a letter before 'U')
• + 12 (from 'NUM' and a letter before 'M')
• + 2 (from 'NUM' and a letter before 'B')
• + 1 (for "NUMBER" itself)

Thus, the rank of the word "NUMBER" is:

360 + 72 + 12 + 2 + 1 = 447

Conclusion

The rank of the word NUMBER when all possible 6-letter words formed from the letters of "NUMBER" are arranged in dictionary order is 447.