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Expert: - 1/4/2011


Question
QUESTION: The letters of the word "BRONZE" are written in all possible order .How many words are possible?.If these words are written out in a dictionary order, what is the rank of the word "BRONZE"

ANSWER: There are 6 letters in BRONZE, and all of them are unique, so there would be 6 ways of choosing the first letter, 5 ways of choosing the 2nd, 4 ways of choosing the 3rd, 3 ways of choosing the 4th, 2 ways of choosing the 5th, and only 1 way of choosing the sixth.
This gives us 6*5*4*3*2*1, or 6 factorial.  This is written as 6!
Now 6*5 is 30, 30*4 is 120, 120*3 is 360, and 360*2 = 720.

Note this value should also be multiplied by 1, but that leaves it the same.


---------- FOLLOW-UP ----------

QUESTION: Thanks a lot for giving answer but what about
If these words are written out in a dictionary order, what is the rank of the word "BRONZE"

Answer
The first word would be the letters in alphabetical order.
That would be BENORZ, then BENOZR, then BENROZ, then BENRZO, etc.

Anyway, there would be 4! ways to arrange the last four letters.
The 2nd letter would be E, N, and O, so 3*4! = 3*72 = 216.

Once R is the 2nd letter, there would be 3! arrangements with E and 3! arrangements with N as the 3rd letter, and then we'd have O as the 3rd letter .  Now 3! = 6, and we'd have two of these, so 2*6 = 12, and 216 + 12 = 228.

The arrangements from here would start BROENZ, then BROEZN, then BRONEZ, and then, finally, BRONZE.  That makes for 4 additional arrangements, so on the 232nd arrangement, the word BRONZE is found.  

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